Why is Hell considered a permanent situation?

There are actually different beliefs on this matter. One issue is differing interpretations of terms like 'unquenchable fire' - does it mean the fire will continually burn the inhabitant or simply that the one thrown in will have no chance of putting out the flames before they are destroyed, and 'second death' - does this mean complete death or only partial (physical) death.

Before you can get to questions about fire, though, is the question of what constitutes Hell at all in the Bible. Most equate the fires of Revelation with Gehenna of the Old Testament and other undesirable descriptions of the afterlife which are considered to be 'Hell.' This also presumes that God changes nothing over the course of Creation. An alternative possibility is that events like the resurrection of Christ and His second coming may alter what happens. For example, consider 1 Peter 3:18-20 says:

For Christ also suffered once for sins, the righteous for the unrighteous, to bring you to God. He was put to death in the body but made alive in the Spirit. After being made alive, he went and made proclamation to the imprisoned spirits — to those who were disobedient long ago when God waited patiently in the days of Noah while the ark was being built. In it only a few people, eight in all, were saved through water.

Thus the short answer is that Hell isn't always considered a permeant solution.

Judicial Process The first verse quoted is progressive. It deals with being held accountable for one's actions (1) by a common judge, (2) then by the high Sanhedrin council, and (3) by the ultimate Judge, God. Only two of these judicial proceedings refer to "our earthly existence." When God judges mankind it will be at the End of the World. (Matthew 13:40-43; 25:31-46) This is when heaven and hell are the final options. [God does not propose to judge a man until after he dies...neither should we! Hebrews 9:27]

Your second verse is an enlargement on what might happen if your conduct and relationship with your adversary (by lawsuit or criminal prosecution) is not resolved in a reasonable and hasty manner. It has nothing to do with the topic of hell, and eternal consequences. It deals with trying to stay out of jail.

These verses have nothing to do with lessening a person's stay in hell by doing any works. The final decision by God is final, and it is based on a person's allegiance to Christ and His righteousness...and grace...and blood!

...so Christ was sacrificed once to take away the sins of many people; and He will appear the second time, not to bear sin, but to bring salvation to those who are waiting for Him. (Hebrews 9:28)

In order to comprehend that the suffering in the lake of fire is everlasting then you need to understand why Jesus resurrects the dead (both the righteous and the unrighteous). After Jesus resurrects all men then eternity has began, either with God in heaven or with the devil in the lake of fire. The form which the resurrected men have is a form that cannot perish but can suffer. You can refer to this verse in Revelation that states the smoke of the condemned souls rises up forever

Revelation 20:14-15

Death and Hades and anyone whose name is not written in the lamb's book of life , were thrown into the lake of fire, the second death.

Also men will seek death so they may flee the wrath of God poured into his cup unmixed but death will flee from them

Revelation 9:6

In those days, men shall seek death and shall not find it

When eternity will start then there would be no more perishing and it will either be everlasting joy and peace in heaven or everlasting torment and gnashing of teeth in hell.

As has already been pointed out in a comment and in one particular answer, there is a danger of mixing up things said by Jesus in chapter 5 of Matthew's gospel, which will lead to misunderstanding. To expand on that, here is the comment in a study Bible re. Mat. 5:22.

"Hell - The Greek word is ge(h)enna, which derives its name from a deep ravine south of Jerusalem, the 'Valley of the (Sons of) Hinnom' (Hebrew ge hinnom). During the reigns of wicked Ahaz and Mannasey, human sacrifices to the Ammonite god Molech were offered there. Josiah desecrated the valley because of the pagan worship there (2 Ki.23:10; see Jer.7:31-32; 19:6). It became a sort of perpetually burning city dump and later a figure for the place of final punishment." NIV Study Bible, p.1420, 1987 edition

When Jesus then gave the account of the beggar, Lazarus, dying and finding himself in bliss (it doesn't say 'heaven', note), and the rich man dying and finding himself in torments, Jesus again used the word 'Hades'. The footnote on Luke 16:23 makes that point, adding:

"Hades is the place to which the wicked dead go to await the final judgment. That torment begins in Hades is evident from the plight of the rich man." (Ibid. p. 1541)

There is something intriguing said about hell and an eternally burning lake of sulphuric fire in Revelation chapter 20. Before the resurrected dead are judged at the great white throne, when Earth and sky have fled from the presence of the one seated on it, we are told that:

"...the devil that deceived them [the nations] was cast into the lake of fire and brimstone, where the beast and the false prophet are, and shall be tormented day and night for ever and ever." Revelation 20:10. A.V.

Then comes the judgement of all humanity, and those whose names are not found in the Book of Life go to join the devil and his agents in that lake of eternal fire. The sequence of events is written like this:

"And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works. And death and hell were cast into the lake of fire. This is the second death." Revelation 20:13-14 A.V.

Death and hell being cast into this lake of sulphuric fire, indicates that death and hell cannot BE that lake of fire. Yet there remains no doubt that the eternal state of those whose names are not found written in the Lamb's Book of Life endure eternal torment. It's not a matter of location; it's about condition, yet the wording here is very particular to show a downward spiral from Hades (hell) to an eternal state from which there can be no release via death, for this part of the Revelation applies to when time has ended, and eternity obtains.

As for Matthew 5:26 - that is a different matter, an illustration showing the need to repent before it is too late for the wrong-doer to avoid judgment. We all have until our deaths to turn repentantly to the Judge of all the Earth, to receive his free pardon, in Christ. But when we die (or, when Christ returns in glory - whichever happens first) the Day of Salvation is, for us, ended, and the Day of Judgment awaits at the Last Day.

Conclusion: The basic "reason why hell is considered to be a permanent situation" is because the Bible clearly teaches permanent torment for those judged as not being in the Lamb's Book of Life ("the smoke of their torment ascendeth up for ever and ever: and they have no rest, day nor night, who worship the beast and his image, and whosoever receiveth the mark of his name" - Rev. 14:11 etc.) But because of inattention to the full scope of Greek words used to deal with the situations of the dead, before and after their resurrection, the English word 'hell' has been used all over the place, when the nuances of the various Greek words ought to have been taken into account. What should be clear, however, is that there is no soul annihilation where no possible awareness of anything exists; the everlasting joy of the blessed dead is contrasted with the everlasting torment of the cursed dead - in the Bible - call that condition what you will.