I have this happen to me on the iPad sometimes, too. I think that what may be causing it is that my aim is off. If I don't directly touch "See step-by-step solution", it doesn't get noticed that I touched it, and the option to see this disappears. I can then only go on to the next question, or if it was the last question in a set, the summary.

I suggest, though, that you use Report A Problem to let Khan Academy know of your problem. It could be that they know of a way to make your choice register.

Also, when you submit your problem report, here are some things to note:

Be sure to do screen prints for this issue. Do one when the "See step-by-step solution" is shown. And then do another one after nothing happens.

When the get back in touch with you, they will tell you to try uninstalling and reinstalling the app. I seriously doubt that uninstalling and reinstalling the app will fix this problem for you. They always make this suggestion, and lately, they have gotten adamant about not going any further with helping you unless you first try this step. It did work for me ONCE on the iPad, and just this week, it worked for me on my Android device (the first time it did the trick on that device, and it was my idea to try it, not Khan's). But usually it does not work. Tell them in your report that you would rather not uninstall and reinstall the app (unless you really are willing to do this).

They will also tell you to clear cookies and cache. These are not options for the iPad app. Remind them in your report that you  won't be able to do this.