Seeing as we will probably be below 11,000 nautical miles, we can use GPS for macro navigation and positioning data. Micro navigation would likely be done with one or more gyros with course correction done by rudder fins and small retro jets. Computers are fast, cheap and small. The only possible problem that we would run into running a solid-state computer in space is shielding. We're probably going to have to sacrifice a bit of weight and get some lead to cover the electronics.
Regular satellites use hardened circuitry, but that's going to be a bit expensive for our purposes.
We'll also forgo the use of a nuclear reactor for power in favor of some lithium batteries. Reentry should really be just a calculation of when to fire the retro jets to force the orbit to decay. After the initial entry where the rocket will get quite hot, we fire a small drag chute and wait until we have 5,000 feet of altitude before firing the main chute. We'll probably want to target an ocean and possibly inflate a balloon to keep the ship from sinking. Then any kind of radio transmission should be sufficient to find the rocket.