The standard preheat fluorescent wiring diagram applies here.
The one difference being that this is manual preheat (press and hold to start). So your switch serves as the starter in the circuit, and also a switch for the supply line. That's why you'll have four wires to the switch, the two for the starter, and the two for mains power.
I presume you attempted to switch this on by pressing, holding, then releasing the switch, correct? That'd be my first question. If the ballast has gone open-circuit or is otherwise faulty, it shouldn't be to much trouble to find a comparable replacement. I'm not sure if those starter switches are readily available, but you could get by with a regular switch and add an automatic starter. You could alternatively install an electronic ballast that doesn't need a starter. That'd require more wires if the ballast was to remain in the base, but should be doable.
The switch could be faulty as well. You could try bypassing one or both of the switch functions. With the both bypassed, the ends of the lamp should glow continuously. If you put the starter portion on a separate switch, nothing should happen when it's open, the ends should glow when closed, then the tube should light normally once opened again.
