How are you going to charge the Belkin Battery Backup unit? If its the common Belkin RG one, the internal SLA battery is only 7.2AH, and if you try to use more than about 70% of its nominal capacity, you'll probably kill it.
Without charging it, starting from fully charged with a good battery, 90mA for four hours a day for a fortnight would be possible - that's about 12 LEDs in four series strings running at 22.5mA. If the battery is old and tired, you wont get more than 2/3 of that
How to wire it up with the parts you have on hand depends on the LED characteristics. At the very least we'd need to know their rated current and their Vf and what current its specified at. The basic idea is multiple series strings of three LEDs to get a total Vf drop of around 9V then add a resistor to limit the current to each individual string. The resistor will probably be several of your 470R ones in parallel to get a suitable value. If you try to use series strings of 4 LEDs without resistors, they'll be bright to start with, but the light output will drop off very quickly as the battery discharges.
Frankly you'd be daft to put it in that light housing - the diffuser just wastes light. You'd do better to tape three or six LEDs together to the end of a piece of solid copper wire you can bend to aim it where you need it, or bounce the light off the shiny side of a crumpled piece of aluminum foil to diffuse it with less loss.