I use the teabag trick. Cut a small piece off of a tea bag (not the top part, but the part that holds the tea b/c it's thinner). Enough to cover the part of the nail that's split. Glue it down w/super glue and then when the super glue dries, file that down gently until it's smooth and cover it w/nail polish. I usually do clear nail polish and then a color so I won't "think" about it.
The teabag fibers act like mesh to keep the nail together and will grow out w/the nail. You might have to redo this if it starts to wear down ... but it usually works. If you can, try to wrap the tea bag under the nail a bit.
ETA -- I think you can go to one of those nail places (the Asian kind) and they can silk wrap the nail for you. That might work.