I am 5'7" and was down to 97 pounds before having my bilateral lung transplant in May. I have put on over 20 pounds since. My doctor had me drinking 4 high protein shakes a day at first then reduced to 3 shakes a day and now I no longer am required to drink them since I am able to eat a full meal now. I understand the concern of gaining too much weight. However, it seems your BMI is still too low and your doctor is probably saying that you need to gain more. Look at serving sizes and try to eat balanced meals.
Due to the meds, I have an issue with my sugar being too high, so I need to try to keep to 3 meals a day. If you don't have the same issue with sugar, you may be able to eat 6 smaller meals a day.
I started walking before leaving the hospital and going up steps (I live in a house with stairs). I am now up to walking a little over 4 miles a day. I am also doing arm exercises. I needed to build back up muscles since I was considered under weight and protein deficient. I am not sure if you have the same issue with your muscles, but if you start out slowly you can build back up the muscles. Between exercising and healing from the surgery, it burns more calories.