Sorry to hear your budgie had such a rough start. It's not easy when it starts like that. I was unlucky myself and brought home a sick budgie (unfortunately he didn't make it), and I did find that he spent all his energy on trying to recover rather than play and have fun. Has a vet checked him again to make sure he's fully recovered? Do you know how old the budgie was when you got him?
I've had two budgies prior to the one I have now. They've all had different interests in terms of what they like and don't like. The one I have now was very young when I got her. She was afraid of everything but very quickly learned to trust me. She hates being anywhere where she can't see, so the bottom of her cage is never in use. She doesn't like things that move either, so she won't play with balls.
To find out what your budgie likes, try leaving a variety of toys around. When you bring your budgie out of the cage, make sure it's easy for her to fly away if there's something she doesn't like to do. Try with a ladder and slowly move her to the bottom and see if she might climb up. Try with a swing and see if you can get her to step up. It can take many tries, and you don't want to keep trying and trying. Set a time limit - maybe 15 minutes. Don't worry if she refuses to do anything - she'll still get used to the training time. Do this a few times a day.
If you have a bell (budgies tend to like noisy things), show her that it'll ring if you move it. Maybe hold her at a distance when you try this so it doesn't scare her.
Everything is new to her, so she needs to learn one little bit at a time. My own budgie was afraid of new things in the beginning, but she's SO curious now.
Good luck with your little budgie