I have a patch of skin on the inside of my elbow that is very white, dry, and rough. It has been here for about 2 months and has not gone away.

Dry skin? If over the elbow it may just be dry skin If over the elbow this is just an extensor surface easily and often presents as dry skin. The other concern would be a plaque of psoriasis. Needs to be seen by a derm doc and maybe biopsied. Best wishes.