So after adding my corn sugar I found that my bottle capper doesn't work so how long can your beer sit in the bottling bucket with the sugar in it before Bottling
How long are you thinking it will be before you get a new one? You could let the yeast ferment out the corn sugar and then start over if you're looking at days.
Not very long. You're exposed to O2 and you're about to restart fermentation. I'd get one ASAP. If you can't get one I'd put a lid on the bucket with an airlock until you get a new capper.