Jews, what does Jewish law say to do with a mentally ill person?

If that person commits a crime?
If that person kills someone?
If that person is put to death/mistreated for their mental illness?

Please give me a source for your answer.
Thank you and a Shayne dank (hope I said that right)
Please point out your source from Haleakala, sorry I was looking for the correct word.
Allinaovas the most extensive answer. Anyone else what to they their hand at answering?
I see that Jewish law is more just than laws where even a 10yo child can be sentenced to death for mistakenly killing someone. Justice cannot be found in things that punish the unknowing person, especially when they did not have malice in their hearts.
Of course G-d is the ultimate judge, no one shall escape that judgment not matter how the trial may turn.

I'm not Jewish
but this might interest you from the New Testament
regarding the feeble minded
1Th 5:14 Now we exhort you, brethren, warn them that are unruly, comfort the feeble minded, support the weak, be patient toward all men.

All countries have civil laws so people generally don't have to pay much attention to Haleakala for such questions.

Jewish law differentiates between a person who has the capacity to understand and one who does not. Thus we find the law treats people differently based on their mental capability- it is also this need to have the ability to understand that exonerates children and why we do not see a child as being capable of sin until after the age of Bar/Bat Mitzi.

When it comes to the mentally ill person, a major place where we learn that they are not culpable is in the laws of capital punishment in Mischa Sanhedrin. In Mischa Makkos where the giving of lashes is discussed, the same point is raised and thus they can be discussed together.

Essentially, before a person can be found guilty of a crime with a physical punishment i.e. capital crimes or lashes, there need to be two witnesses. These witnesses must not only have observed the crime, but must have warned the person beforehand. More than that, not only must the person have been warned, but they must have acknowledged the warning and then gone on to do the act.

This is where issue of mental capacity is raised. If the mentally ill person had the capacity to understand the warning at the time, and the cap city to meaningfully answer and acknowledge the warning, then they have the capacity to be found guilty for their crime. However, a mentally ill person who lacks this capacity cannot be said to have received a warning since they did not have the capacity to understand it. Thus, they did not receive a warning as it was not understandable to them- just as if the warning is given in a language the person does not understand it is as if no warning was given! Thus for such a mentally ill person we cannot use capital punishment or any physical punishment.

When it comes to their general well being, they should be treated and cared for, hopefully to the point they recover and are well again. Rambam makes this point in his works, both in his medical writings and his religious writings where he discards the idea of possession/spirits for mental illness and insists on medical treatment for it.

Wow! I was going to answer, however I saw the Talmid Chakham's answer (user Allonyoav) and while there is much to add I will hold back from doing so out of respect for him.