Gia,
I think there's two parts here. One, you can't get rid of your mom. Secondly, you have her junk. Mom being in a nursing home should make things much more manageable emotionally albeit you are still her main caretaker. But it is not 24-7. There are boundaries you can set and it sounds like you have which is good.
As for her stuff, that may be much more triggering that you think. Even if you think something isn't a big deal compared to other things, it still is a deal and if it isn't big, why not just take care of it? As Ms Spock said, little by little even if it's hard. It's not just about you and your mom, but you and your mom and her stuff that you are holding onto for her. Even if she was gone forever, you still have her things to deal with. It is amazing how people are tied to things. I think you just need to analyze what is really burdening you and start with what you can control/handle. Example, do you really need to keep that 12 set dining table your mom can't depart with? Must you keep in storage her teddy bears from her childhood? I don't know what she has but it seems some it may not be necessary for her or your sanity.
I'm sorry to hear that your husband is sick as well. You are carrying alot - unloading every little ounce helps.