Recently purchased a 10 stop 100mm ND filter and am using it with the LEE mounting kit on a 10-20mm wide angle lens. When I shoot long exposures, a faint reflection of the camera lens can be seen in the shot. This only happens when I expose for longer times (1 - 2 minutes). Is there anything I can do to prevent this from happening?
Filters with anti-reflective coatings will help reduce this sort of thing. Filters with multiple coatings (on both front and back surfaces) will reduce it more. You cannot eliminate it entirely (well, you could if all your "lenses" were mirrors). I think even the best multi-coated optics still have something like 0.02% reflectance. If the subject you're shooting is dark enough or you're using enough darkening filters, the reflected light can become a significant fraction of the image, and shows up as flare.
Incidentally, you can create a variable neutral density filter with two polarized filters (a circular polarizer and a linear polarizer). Mount the circular polarizer closest to the camera, and the linear polarizer in front. As you rotate the two relative to each other, the light reduction will vary from 50% to 100%. Of course now you've got 4 filter surfaces (2 per filter - front and back) causing reflections instead of just 2 from a single filter. But it can be cheaper to get two polarizers with good multi-coatings, instead of a half dozen ND filters with multi-coatings.
The only other way to eliminate it is to shield the lens/filter from the bright object causing the flare. But in your photo the light seems to be coming from the entire background. Still, on the off-chance it's light coming in from outside the image, you can use a lens hood to block out stray light. If there are any light gaps between your lens and filter mounting system, try covering that with tape. Better (more expensive) lenses and fixed focal length lenses will also have internal baffles to help reduce this sort of flare. Zooms tend to get more of it due to having more elements and their variable position reducing the effectiveness of baffling. So try a fixed focus lens instead.