How is one supposed to "know" whether or not the Book of Mormon is true?

Mormons don't "simply ask" people to pray about it, but to read the book, to make knowing whether it is true or not the focus of their desires, and then to pray and ask God, and "ponder it in their hearts", and if he has "a sincere heart, with real intent, having faith in Christ" (Moroni 10:3-5), he will know whether or not it is true.

Simply, because historical and archaeological sources are the wisdom of man, but the revelations of God can penetrate the heart. Testimonies of the Book of Mormon come just as Paul, in Galatians 1:11-12, received a testimony of the gospel and learned its truths:

11 But I certify you, brethren, that the gospel which was preached of me is not after man.

12 For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.

It is by the Spirit of God that one gains a true testimony. This can happen in any number of ways, according to God's will and the person's humility. Mormons don't really limit how God will answer another person's prayers. But almost universally, a feeling of peace will enter into their heart and mind.

As an aside, I think I understand why you included "prophetical" in your list along with "historical" and "archaeological" fields, in that you probably know that Mormons tend to discourage looking at Bible references for proof that the Book of Mormon is true. I think that is the difference. When one looks for "proof," their heart is not in the right place. When one seeks a divine witness, however, they are more ready for the sacred revelations of God.

Of course, this is not to say that scriptures should not be used as a reference. That would be foolish. Use the scriptures, study them, and use them to reinforce your own understanding, and enlighten your mind, and avoid deception.

Here's an official page describing how to know the Book of Mormon is true (scroll down).

Again, the best thing one can do to know whether it's true or not is to read the book with a sincere heart, with real intent, having faith in Christ. It is not merely a matter of praying about it.

Paul answers that question rather soundly: See Galatians 1:6-9

⁶ I marvel that ye are so soon removed from him that called you into the grace of Christ unto another gospel: ⁷ Which is not another; but there be some that trouble you, and would pervert the gospel of Christ. ⁸ But though we, or an angel from heaven, preach any other gospel unto you than that which we have preached unto you, let him be accursed. ⁹ As we said before, so say I now again, if any man preach any other gospel unto you than that ye have received, let him be accursed.

Emphasis Mine

LDS teaching regarding seeking a divine confirmation of something frequently references the following passages:

Moroni 10:3 Behold, I would exhort you that when ye shall read these things, if it be wisdom in God that ye should read them, that ye would remember how merciful the Lord hath been unto the children of men, from the creation of Adam even down until the time that ye shall receive these things, and ponder it in your hearts.

This implies an examination of Biblical teaching with an attitude of worship rather than skepticism, and is before the cited injunction to pray.

Doctrine and Covenants 8:2 8:2 Yea, behold, I will tell you in your mind and in your heart, by the Holy Ghost, which shall come upon you and which shall dwell in your heart.

Doctrine and Covenants 9:7-8 7 Behold, you have not understood; you have supposed that I would give it unto you, when you took no thought save it was to ask me. 8 But, behold, I say unto you, that you must study it out in your mind; then you must ask me if it be right, and if it is right I will cause that your bosom shall burn within you; therefore, you shall feel that it is right.

All these imply some degree of intellectual effort, not excluding examination of the historical, archaeological, and prophetical evidence available, but answers to prayer are matters of faith, not intellectual conviction alone.

It is not possible to specify exactly how a person will recognize a divine confirmation, because it may be unique to the individual.

Doctrine and Covenants 46:11-14 11 For all have not every gift given unto them; for there are many gifts, and to every man is given a gift by the Spirit of God. 12 To some is given one, and to some is given another, that all may be profited thereby. 13 To some it is given by the Holy Ghost to know that Jesus Christ is the Son of God, and that he was crucified for the sins of the world. 14 To others it is given to believe on their words, that they also might have eternal life if they continue faithful.

Believers report a wide variety of experiences.

According to LDS history, Joseph Smith was meditating on James 1:5 when he had his first encounter with Jesus. That verse says:

If any of you lacks wisdom, you should ask God, who gives generously to all without finding fault, and it will be given to you.

That guiding principle has underscored much of Mormon theology since - God gives wisdom to those who ask for it. As Mormons believe their scriptures to be wisdom from God, then asking God for the wisdom to see that is a natural extension.

This Christianity blog post by a certain Affable Geek may help with more insight on that meditation.