Victory housing has respite care.

Some nursing homes accept people for short term visits. For example, several friends used Powhatan Nursing Home off of Powhatan in McLean for weekend or 1-2 week stays for their parents. I do not know the rates, but it helped them have a respite or go on vacations with their children.

I know it looks rather shabby, but it has a great reputation.

OP - I would suggest sharing your concerns directly with your sister and mapping out a monthly weekend, but include half of the long holiday weekends so that you all might get away if you like.
Also it seems fair to plan to have at least one week of vacation without your mother for your family, if not two during the course of a year. 70 is not that old and assuming your child is older now, your Mom 's personality might improve if she has other social connections.
You might also consider trying to introduce your Mother to a local senior center or public library to see if she might enjoy activities there or meet others to have connections outside of your family. See if you can get her qualified to use paratransit for those with a disability/seniors so she can get around if she does not drive.