How do you make her trust you after an affair?

You need to make her believe you could never risk it again because you need her more than anything in the world, and all without being clingy. Buy her some things she likes without going overboard, plan to spend alone time with her so she's all you're focused on at that moment, and let her know what you're up to from time to time without her even having to ask; show her that you're serious about winning back her trust.