I've owned a KeepCup for about a year and though I clean it after use it really is starting to stink.
I've tried scrubbing it with a scouring pad and detergent and leaving it to soak overnight, but it still smells bad.
The cup looks like the one below. All pieces can be disassembled to make cleaning easy. It's like the smell is in the plastic it's self.
Wash it in a dishwasher once every month or two. I own a Contigo mug, and once had similar problems with the plastic/rubber lid. I found some tips on their website, but ultimately found that if I washed the mug and lid in the dishwasher about once a month, the smell and staining would go away. I don't have a dishwasher, but find that I visit someone who does often enough to keep the smell at bay.
KeepCup also includes cleaning tips on their website, which you may want to try first if you don't have access to a dishwasher.
Alternatively, KeepCup offers a one year warranty, so if you've owned the cup for less than a year you may want to try contacting them.