Weights after Anorexia?

So I am 4 months into my recovery from Anorexia which almost claimed my life and as a result I lost a lot of muscle and bone density and mass. Now that I am in a healthy weight range and have successfully been re-fed and gaining weight my doctor is now telling me that I need to get into a strength routine with weights/body weights to build muscle again as I have re-gained weight but not enough (In her opinion) of muscle/muscle mass.

I am curious as to what you all suggest? I have put together a plan that I would like to attempt today to start this regiment of strength training:

3 sets of 8-12 reps for each exercise

-Bodyweight squats
-Bodyweight Lunges
-Step Ups with dumbbells
-Bench Press
-Bicep Curls
-Military/Overhead press
- Bent Over Dumbbell rows
Lat Pull Downs
Bodyweight push ups (I have to start while in a semi-push up position with my knees down on the ground as I lack arm strength but would like to eventually start doing full push ups as my strength comes back)

I am planning on doing that as it was suggested I do 2-4 exercises for the leg muscle, 1 for chest, 1 for back, 1 for biceps and 1 for triceps.

I am going to also try power walking on the treadmill on a raised incline as I believe that is good for building leg muscles.

I am also hoping this helps my physical appearance because during recovery I gained back my weight and a few extra but am not worried about m weight. I would like to develop a more toned/healthy look as right now I am pretty much the definition of skinny fat. I lack a lot of muscle and there for definition. My tummy and thighs are especially undertoned. I am not looking to change my weight so I am keeping my calories high and eat a complex carb and source of protein at every meal (Usually 1 cup serving of rice, a side of vegetables and a piece of chicken or a cut of beef/steak) I do eat treats once and a while but try and eat as clean as possible otherwise and am now drinking protein shakes.

Besides for doing this weight training regiment 3x's per week I will also do cardio 4-5x's a week. I am hoping that the combonation of cardio with weights will help me to both gain muscle but also reduce some of the fat % on my body without losing any weight.

I am wondering your thoguths and if it would be an alright idea if I started out at first with no/very light weights so I can focus on proper forum? Will I still gain muscle with bodyweight exercises?