No need to do anything. Just allow carbonation to occur you should get the same result that many brewers before you have done. For many years, brewers have put sugar directly into the bottle dry - without boiling - and had good beer come out. No boiling, nothing else. That's why home brew shops sell sugar measures and funnels.
Personally, I believe it is better to boil and sanitise the sugar first. I would not be concerned if I were you as what has happened has already happened. If it's infected, that's already happened. If the beer is fine, that has already happened to.
I think you'll be ok