Suzanne--
I would recommend taking taxis and staying where ever you want. Taxis are very, very affordable and plentiful in Madrid and your mother will get lots of exercise just standing and walking in the museums. For someone who isn't used to a lot of walking, just a few blocks from the hotel to the museum could be quite tiring, especially in hilly Madrid. Avoid the metro, it is full of stairs, which could be very hard for an older person who isn't used to them.