Great theme and lovely colours. In unity there's a gameobject called the EventSystem, and you can set what it's currently targetting for UI, so you can set the text field used for typing in. Personally I like that you have to type to move the pieces, as it makes it a bit of a typing game as well as a puzzle, I think once you evaluate the move after the user presses enter, you should clear the text field.
Awesome job for your first jam, my first jam wasn't nearly as good.