Hi, Liz ~
Welcome. A 5 cm AN is considered quite large and post-op recovery is necessarily going to be a bit of a struggle. Many people think that this kind of surgery is akin to having a gallbladder removed. You go in for the surgery, it happens while you're under the anesthesia then you lay in a hospital bed for a few days, slowly come around and eventually, you're discharged. Then you're home, recuperating, taking it easy and within a few weeks, you're 'good as new'. Not so - for most AN post-op patients, anyway.
Your son's shaky legs are probably due in part to his brain's need to re-learn how to balance and in part due to inactivity. Being tired (fatigued, really) is very common, post-op. The operation is long and complicated and takes a big toll on the body. Fortunately, your son does have his youth on his side and should make a full recovery, in time. How much time? I have no earthly idea. The recovery process varies wildly but 5 weeks is not that long and he should see improvement soon. Not totally normalcy, mind you, but less fatigue as his body heals. I would definitely consult your doctor regarding the depression. There are many medications available that can alleviate that condition and he should pursue them, under a doctor's supervision, of course.
I would definitely suggest that your son attempt to walk as much as possible. Our bodies were made to move and even something as simple as walking can help wobbly legs gain more flexibility and strength. Very short walks (to the mailbox, for instance) are a good beginning. Rest as needed, afterward. Slowly increase the walking as he feels able. At 25, your son has a good chance of making a solid recovery but he will have to fight for it, as we all do. I know you'll help him and so will the folks on this forum because we've been there and can understand his struggle. Please feel free to ask whatever question arises. We aren't doctors but we have a lot of experience dealing with AN issues and our members are eager to help. I hope these posts turn out to be useful for your son's recovery.
Jim