You can use the existing chainset; take off the big ring, replace the chainring bolts with short, track, ones and, if necessary, put the ring on the inside of the arms to get a good chain line. That's what I have done with every fixed gear I have ever ridden. No need to use a 1/8th track chain either so you can probably use the chain you have if it is not too worn, just shorten it. You can easily find a good ratio with a 39 ring (assuming your inner ring is 39?). Cheap, quick and effective.
You can use the existing chainset; take off the big ring, replace the chainring bolts with short, track, ones and, if necessary, put the ring on the inside of the arms to get a good chain line. That's what I have done with every fixed gear I have ever ridden. No need to use a 1/8th track chain either so you can probably use the chain you have if it is not too worn, just shorten it. You can easily find a good ratio with a 39 ring (assuming your inner ring is 39?). Cheap, quick and effective.