Subwoofer

Given your limitations and speakers it might make sense to try the KEF offerings as they have settings to support non-optimal placement. The KC 62 might be a bit small for that room, the KF92 might be a better choice. 
 

I have LS50’s, and one sub is better than no sub. Yes, two might be better, but not required to improve your sound. 
 

One sub is very hard to put in a musical synergy with the speakers...

But if someone want only boom boom sound , it is very easy to plug it...

I plug mine few minutes and sold it... 😁 because i could not think how i would make it "musically integrated" ... i think that it must takes 2 ...

A balance sound is not a boom boom sound added...

But my short experience with ONE sub  ( Kreisel K9 ) was not happy and i could not figure out a way to integrate it "musically"  ...

But once this is said i am not a sub woofer specialist at all...But i love music and i hate boom boom... Perhaps i am wrong and there is a way in some room to do it musically with ONE sub... We must ask to a sub-woofer specialist for confirmation or debunking ...My intuition is location and room dimensions are very important for this problem then there is no absolute rule and in some case one sub can do the job perhaps... In most general case i think it takes more than one...

@jpwarren58 

I also have KEF speakers and wanted a subwoofer. My solution was to put an SVS 2000 Pro on the back wall, about six feet from my listening spot.  Their software is great and it sounds seamless.

All the best.

I have two subs, but typically only keep one hooked up mainly due to WAF, and I also have very limited placement options. My room is 20x 24 x9, and opens into another large area with a vaulted ceiling, so there’s some cubic footage in play. I prefer both subs hooked up, but one sounds better in my situation than none. It’s a treat when I run both subs and pull the speakers out where they sound best. You can buy a decent sub from Dayton or Monoprice for < $200 if you want to experiment before sinking more money into it.

Even low cost subs like the Dayton have variable low pass crossover frequency options, separate gain, and two phase options that help give some options to dial in the bass to suit your needs, even without many placement options. I run mine from the amps into the high level inputs, keep the low pass crossover frequency very low to avoid directionality and resonance issues, and keep the gain very low so it just augments the bass from the mains. I don’t "feature" the sub. It's subtle... I barely hear it except for on heavier bass passages.

I have a KEF KF92 subwoofer in a room similar to yours. I also got their remote transmitter for it so I can locate it where it sounds best, which for me ended up being in the middle of the room, behind the sofa. This has different curve settings for different locations in a room: Corner, wall, room. Good starting points. 

This is my third sub, and I am very happy with it compared to the others. I see that someone recommended the SVS 3000 Micro. I had a pair of them for a short while, then sold them. They didn't go very low in my room and I couldn't get a good setup with them. Though I will say SVS is a good company and the fit and finish of the Micros was excellent.