Use a V shaped block to hold em, put tape all round where you are going to drill then mark the hole with a sharp centre pop. Using a new, sharp as possible, small drill (say 3-5mm), drill a pilot hole. It's a lot easier with a pillar drill. If you don't fancy spending a few extra quid buying a stand to convert the Black and Decker to a pillar and you need to make the V-blocks, can you knock up a U-shaped channel you can use to guide the drill body? Beats trying to hold it square and push when it's lined up.
Plan B is to make a shield/guide out of an old nut and a length of pipe (or just thick wall pipe). This is safer but more work. The nut is welded or epoxied to the pipe. You drill a hole through the nut and pipe. You can get this wrong a lot of times before it gets that expensive, but it's easier to just open up the thread in the nut that start a new hole. Next you slide the pipe over the expensive to replace handlebars and clamp in place and/or push on over layers of tape. Use the hole in the nut and pipe to guide the drill.
Anyway, you don't want to listen to me, I once helped destroyed a very expensive truck axle by slipping with the drill. Oil coiming out of the hole you just drilled is a bad sign

Andy