Determining a good maxim is an art. I'd think along these lines: if home care really is too much for me, then the result hurts my mother. Under that condition, and there being no other alternatives, then I think the nursing home is appropriate (assuming the nursing home is itself good). That is, the promise cannot reasonably be kept.

Of course it matters just how home care is "too much for me." But then you have to add detail to the initial givens.

As to the imperative of the promise, In Metaphysics of Morals, Kant makes explicit that in the case of competing imperatives, one rules and the others lose their force.

In the light of Tim Wood's good response, it looks like there are no easy outs. You promised the impossible; you are not the first person to do that. Impossible promises can not be kept.

A better promise would have been, "Mother, I promise to take care of you as long as I can, and then I will have to find another way for you to be cared for." Or, maybe you can't take care of mother. Then, "I love you mother, but I can am not strong enough to lift you. You have to live in a nursing home." (And in many places in the world, there is no nursing home available...)

People usually do the best they can. That may not be as good as they imagine they should do, but be realistic.