How old is the converter? I assume the converter is still charging the battery
since otherwise your battery
would be dead as door nail in not very much time. My best guess is that the electrolytic capacitors in the converter have dried up. The capacitors are the large cylinders. Carefully unsolder them and simply order components with same capacitance and voltage rating and approximate package size from Digikey. The specifications are written on the packaging. Remember to mark which contact was marked with a stripe since that is ground. If you put them in backward they have a habit of lighting
on fire. Shouldn't cost you more than $5 bucks+shipping.
If that is not the problem or you are scared by electronics simply buy a new converter. A new converter will be safer and will save you money by treating your battery
better nicely. Size your converter based what you are using in terms of 12 V power. If it just the lights
, switch to LED and buy a small cheap battery charger.
Consider switching to LED lights
and a solar panel
unless you are using a lot of other electrical