The Android version works perfectly fine with touch controls. The UI is slightly different for PC and might be less practical with touch controls but it should still work.
As far as I can tell everything is setup right. You might need to reload the purchase page. There should be a button. If it's still not working let me know.
Would be nice to be able to keep the button down to move up instead of clicking like crazy. Being able to use the space bar might be nice too. Congrats on your first game!