Congrats on your conversion! A coaster brake is a far better set-up than a fixtie in my opinion. To take up the extra space you have two options, both of which seem obvious if you think about it. 1- you can bend the frame to a narrower configuation,,,, wich is actually fairly easy on some frames and bleeping difficult on others. or 2- you can use washers / spacers to take up the gap, IF your axle is long enough.
Personally, I prefer bending the frame. The last time I did it, I rigged a device consisting of 2 2x4's and a piece of threaded rod to make a clamp to pull the stays together evenly. If you rig such a device, pull the stays together about 2/3rds of the way to the drop outs, not at the very end so as not to damage them. Just tighten the threaded rod until the frame is about right, then loosen it to check. You'll have some springback, so you'll wind up over tightening it a little. I've used that technique on a number of steel frames, and one aluminum one..... and it always works fine.