Years ago I used to do minor tire repair on cars. A rim leak was almost always caused by corrosion or debris on the rim and the bead seal. Our solution was to:
1) Mark the location of the valve stem and weights on the tire.
2) Remove the tire.
3) Use a drill with a wire brush attachment and thoroughly clean the area.
4) Reinstall the tire aligning the valve stem to the previous mark.
5) Water test for leaks
6) Reinstall any weights
You probably have a similar situation with some debris or corrosion. Here's my recommendation:
1) Remove the wheel from the bike and take it to a tire store. Place it in their water tank and mark the location of all leaks (over pressuring the tire will help).
2) Break the seal of the bead and rim and examine the rim. You can then determine if you need to remove the tire or not.
3) If you're lucky, just remove the debris and reseat the tire.
4) If your not lucky and there is corrosion, remove the tire and clean thoroughly.
5) In both cases, place the wheel back in the water tank to make sure that the leaks are gone.
6) If the corrosion is bad enough, this will only be a stop gap. Start looking for a new wheel. You will probably have to replace it at some point.
7) Once you determine that the rim is no longer leaking, reinstall the wheel on the bike and ride on.
Speaking of Ride On, that is another possible solution. If after water tanking the wheel you can see that the leak is minor and can find no debris or corrosion. Throw a bottle and a half of Ride on in the tire, remove any weights and enjoy the ride.
Another possible solution is to put a tube in it and call it good.
While you're doing all of this and have the tire apart, get a right angle valve stem and install it. It make putting air in the tire way easier.