How well will the transplanted kidney work with the old one?

Old one not function. Typically when someone is given a kidney transplant it is because both kidneys together are not working adequately. When the transplant is put in it is attached to the blood vessels that the previous organ was using. It does not cause any problems for a non-functioning organ to be left in place.
Old & new kidney(s) If you had significant function in one of your kidneys, you probably would not need a kidney transplant. Most of the times, when the function of your kidneys are down to less than 10-15% of normal, you will benefit from having a transplant.
Very well. Although one can survive with only one kidney; transplant patients do quite well and have a return to normal renal(kidney) function. It is usually the immunosuppressant medications used to prevent rejection of the new organ that can be the most problematic.