Stainless I hate it. Some varieties work-harden if you breath on it.
Now you've tried and failed thus creating a hard spot, an extra hard drill as advised above will be essential, and even they might struggle.
Otherwise, if you can start again where the stainless is still virgin, an ordinary sharp drill has a good chance provided you attack vigorously from the get go with lots of pressure and some cutting fluid. If you can keep the drill cutting all will be well, but any hesitation or insufficient pressure, and the tool rubs, hardens the steel, which promptly blunts the drill. Work-hardened stainless is harder than most materials, which is why they make knives and scalpels out of it.
You stand a better chance with a pricey drill in the first place.
Some stainless steels machine well, they're not all evil.
Dave