You can buy a proper Surly LHT for what the Rholoff costs, as I imagine you know.
To have the sexiest set-up, with straight chain, and no brace to the chainstay, you need a very rugged tight fitting horizontal drop. Otherwise, working the hub will just twist it within the drop , the axle can spin in there. You can use the brace, and that should work with your drop ("slop" since it's sideways?) but sometimes it torks the stay moreso than they were designed to deal with.
My guess would be that your stay is probably strong enough for a brace (obviously depends on your level of use), and your horizontal is probably able to adjust the chain. But you can't get away with the unbraced model:
"Does anyone know the rear spacing on this bike?"
Normally single speeds are narrower than the MTB spacing used by the Rohloff, which is 135. You can measure the bike you have by just measuring between the plates. Mark up a popscicle stick in there, and then measure the marks, or take the rear wheel off.