Legally speaking, 0, since the Outer Space Treaty forbids placing nuclear weapons and other weapons of mass destruction in orbit, and all spacefaring nations so far have signed. Even if the military kept things under wraps, I'm not so sure that stationing nukes in orbit would provide a significant advantage, as you would only be able to strike a particular target when it rotates under your orbit, which means you can hit a target only twice a day at fairly specific times, while ICBMs can be launched at any time. In addition, modern ICBMs have flight times on the order of 30 minutes ([0], [1]), so satellites aren't going to be reaching a target much faster.
Edit: forgot to address another point
>They should be detectable, but on the other hand there are so many satellites that it'd be hard to index them all.
We already track more than half a million pieces of space debris, with 20,000 of those larger than a softball ([2]), so I'd imagine a "secret" satellite wouldn't stay secret for long.
[0]: https://en.wikipedia.org/wiki/Intercontinental_ballistic_missile#Flight_phases
[1]: https://space.stackexchange.com/questions/12028/is-it-correct-that-it-takes-approx-30-minutes-for-an-icbm-to-reach-russia
[2]: https://www.nasa.gov/mission_pages/station/news/orbital_debris.html