Your mother-in-law can only move to the UK with you if she gets an
Elderly Dependent Relative visa. For an Elderly Dependent Relative visa, she needs to prove that she is entirely financially dependent on you and cannot survive in the US alone. She also needs to prove that she has no other relatives in the US who would be able take care of her (and therefore that she has no choice but to live with you because she will not be able survive in the US without you).
This is a difficult visa to get because the majority of inlaws are not completely financially dependent on the child that is living in the UK and they do also have other relatives in the US that can take care of them. I've been a member of this forum for 4 years now and I don't think I've ever actually seen anyone manage to get one before. If she does manage to get one though, she will be eligible to use the NHS.
The only other visa she might be able to get is a 6-month
family visitor visa, but she will not be allowed to stay in the UK for more than 6 months and it will be illegal for her to look after the children while she is there (because that is considered to be 'work' and work is illegal as a visitor). She also will not be allowed to use the NHS.
Ultimately she may apply for ILR, would this be a better option?
If she can get an Elderly Dependent Relative visa then she will automatically be granted permanent residence (ILE) in the UK, so she won't need to apply for ILR as well because she will already have it (ILE is Indefinite Leave to Enter and ILR is Indefinite Leave to Remain - they are the same visa, but ILE is granted outside the UK and ILR is granted inside the UK).
However, if she comes to the UK on a family visitor visa she will have to leave the UK within 6 months and will not qualify for ILE or ILR at all.
