A lot of people say Sports Physical Therapists (Sports PTs) and Certified Athletic Trainers (ATCs) are better at sports needs better the normal PTs and uncertified Athletic Trainers. Many college athletic programs want people with double certifications - Sports PT/ATC, like the guy I went to, but there aren't many like that. A PT or ATC who was also a skating coach might be good too. Regardless, ask other athletes for someone good.
I have started to use the weight machines my pool center keeps in a back room. I'm so pleased with what the weight machines can do for me, that ordinary exercises simply don't. A lot of sports experts say they are outmoded, and that you should exercise only by doing the things the way you do in the sports themselves, but at least for me, machines help a lot. In a few weeks I almost doubled the amount I can use on most of the machines, including leg presses. Decades of non-machine exercises didn't have a similar effect, for me.
Experts are good, but personal results are better.
Weight machines keep you body aligned right, so if you are struggling at first to do something, they won't push you out of line, the way a body weight exercise can. The leg press machine (which exercises a lot of what you are using when you rise on one leg) is normally used with both legs. You can use one leg, though it is a bit awkward. Machines have to be adjusted right so you aren't using your body in weird ways - if you find a gym with a trainer, or someone else is knowledgeable, they can help a lot. You need to be comfortable lifting your own weight (or twice your weight with two legs, with each leg supporting 1/2 the weight) to get up at all from knee bends. To jump you need more.
Christi Yamaguchi [sp?] says you should be able to leg press 3 times your weight (I guess with both legs) in her book. (Does that mean 3X for triple jumps, 4X for quads? I'm only pushing 2X.)
Some skating moves require you go beyond right angle knees, but be careful, and don't push the weight limits harder than you need to for that. Talk to your Sports PT or ATC first, because I think deep knee bends are part of what they worry about for knees.
Some books say that weights help because you can force yourself to use as much strength as you safely can, whereas body weight exercises are limited by body weight, and that doing some exercises that push your strength to its limits is important.
My PT/ATC said it is best to use all your muscles that you can. E.g., you can do leg presses with just your quads, but that is both inefficient, and bad for body alignment (including knees) when you do it off the machine. Conscript every possible core muscle you can - e.g., gluts, obliques, abdominals, thighs including psoas, lower back, etc., to keep knee motion centered right. A really good sports PT or ATC is extremely helpful - most people can't sense when all the muscles possible are in use. The Sports PTs and ATCs know how to tell.