How is "abstain from things strangled" not contradictory to "Not that which goeth into the mouth defileth a man"?

The Law was meant to cause people to realize they are sinful and unable to fulfill God's will perfectly [1], so that when Christ comes, they would see Him as the Liberator [2] from the sense of condemnation that the Law had given them, because He is teaching them that it's all in one's mind [3], and that we can, like him, simply not judge [4] what we eat, and get away with it.

So, the two verses juxtaposed in the question for the comparison do not contradict each other since, again, both are parts of a logical sequence of events, namely the explained progression from God who condemns to God who liberates.

_{[1] "If the ministry that brought condemnation was glorious, how much more glorious is the ministry that brings righteousness!" (2 Cor. 3:9).
[2] Jesus Christ liberator: a critical Christology for our time. Leonardo Boff. 1978. p. 67.
[3] "So this weak brother or sister, for whom Christ died, is destroyed by your knowledge" (1 Cor. 8:11); "What goes into someone’s mouth does not defile them, but what comes out of their mouth, that is what defiles them" (Matt. 15:11).
[4] "I am referring to the other person's conscience, not yours. For why is my freedom being judged by another's conscience?" (1 Cor. 10:29); "Do not judge, and you will not be judged. Do not condemn, and you will not be condemned. Forgive, and you will be forgiven" (Luke 6:37).}

One way to come at this is not to treat it mathematically. When we treat natural language mathematically we are inclined to say none must mean 0% and all must mean 100%. However often when we use natural language there can be exceptions to statements (i.e. an implicit 'some' or other limitation on the statement). Such as when one says, "but everyone is saying it." It is unlikely that everyone thinks or says the same thing. Unfortunately I am not a linguist so the precise explanation for how natural language differs from mathematical language is currently beyond me.

Another way at this, from a scriptural viewpoint is to consider what Paul later wrote: "All things are lawful for me, but not all things are helpful." In light of this we could state that Jesus is saying it is lawful to eat any food. Later, caring for the flock, the apostles give guidance not to eat certain foods because, though not sinful, they may have a negative spiritual effect on the eater.

The foods in question are:

1. Food which has been offered to idols
2. Blood
3. Animals which have been strangled

Food which has been offered to idols ought to be avoided in case it causes problems for people of weak faith. There's a good article which addresses this question here. As the article points out (based on this scripture), there is actually no question of sin about eating such food, in itself, but one ought not to cause a brother or sister to stumble. This is particularly important when dealing with people that have a background where they offered food to idols and they are sensitive to it.

As for blood. Moses wrote not to eat it. But Jesus and Paul both affirmed that what we eat does not make us sinful. Here I am afraid I don't know the answer. However, I believe it involves the Christian being under grace, and not under law. The caveat is that there is a reason given with the command not to eat blood, which is that the life is in it. Even if the law has been superseded by the law of the Spirit, the underlying reason has not changed. So one might, along with the apostles, abstain from eating blood out of prudence.

As for eating strangled animals I am drawn in two directions: firstly, to apply the principle established in scripture that all foods are clean and then to consider that perhaps the apostles had some concern for us that I don't know of.

The second is to recall the story of Samson, the Nazirite. The nazirite vow included not going near dead bodies. However Samson had no regard for this, and touched a dead body. The LORD is slow to anger. But eventually Samson suffered greatly for not living as a Nazirite.

The reason I bring up this story about Samson is that we have a command from the apostles, given in the Spirit. I think that the reason they gave it was out of care for the flock. Like a mother's or father's commands these are not law, per se, but perhaps a graceful garland to be worn.

The context of Jesus' words are about ritual purity through obeying the Jewish law. Jesus is saying that there are no kinds of food which are intrinsically sinful, and that focusing on the food misses the point of those laws: that it is a person's heart which defiles them. Our obedience to the law shows the state of our hearts, but does not determine them.

But in the new Law of Christ God looks for faithfulness through other means. The time of the Levitical Law is fulfilled and God no longer considers any foods more or less holy than any others. This is how I understand Acts 10.

But just because pork and shellfish are no longer forbidden does not mean that there are no other ways to eat sinfully. Gluttony, selfishness, drunkenness are all said in the NT to be wrong. And in Acts 15 the issue of pagan sacrifices are raised. The criteria given by the council are largely reminiscent of the law given to Noah, with the addition of sacrifices. Christians are not to participate in pagan religious ceremonies. Paul discusses this in more detail in 1 Corinthians 8 and says that even food offered to idols is not intrinsically sinful, and we should make our decision based on what will help our fellow believers. If our eating of the food would cause some to think Christianity and Paganism are compatible, then we should not eat it. If they don't, then go ahead.

In short: no food is intrinsically off-limits, but other factors will at times inform our decisions.