G.,
It would be a good idea to speak with an attorney. Laws do vary from state to state, but there are certain things that tend to be universal.
You say that both of their names are on the lease. Are they the only two?
I am hoping that you and your husband are NOT on that lease. If your MIL and her brother are the only two people who signed the contract for the lease, then they (and their estates upon their deaths) are the only people responsible.
If you and/or your husband are not part of that contract, DO NOT pay the landlord. He would have no legal right to collect from you (as long as you are not part of that contract).
If your MIL and her brother break the lease, the landlord can certainly try to go after them for the next three month's rent. However, as their limited funds go directly to their respective care facilities, he will have very little luck with that. You know that saying 'you can't get blood from a turnip,' right?
While he could attempt legal action to collect rent for the next three months, knowing that there is little chance of him getting any real financial compensation, he may decide it is not worth his and his attorney's time.
The big unknown is her brother's complete financial picture. If he does have any assets, the landlord may decide it may be worth it to pursue action.
Bottom line for you: IF you and husband are not on that lease, you are not responsible. Don't pay this guy, and don't become frightened by threats of legal action. DO check with an attorney, particularly since you don't know the complete financial picture of your MIL's brother. Generally a well-worded letter from an attorney shuts down those threats very quickly.
Good luck and take care of yourself. Being a care-giver and over-seeing elderly relatives' lives can be so difficult, both emotionally and physically.
J. F.