I get inflamed gums from my braces all the time. I went to my dentist for it but she said my teeth and gums are healthy and didn't really help. I tried lots of different swollen gum treatments but the one that works best is rinsing with warm salt water. Salt water kills bacteria and has anti-inflammatory properties.
When I notice that my gums are swollen, I add lots of salt to warm water and stir until it dissolves. Then, I rinse my mouth with this water several times a day. The swelling usually goes down by the end of the day, or at most in two days.