Recovery is just as important as everything else if not more so. Sleep is an issue with me and tbh I don't know the answer to that one. I stopped drinking coffee a fortnight ago and although I've been feeling a lot less anxious than before and lost the notion for alcohol (I suspect that's all related), I still lose two or three nights' sleep a week, which is disappointing, although I spend far too long on the laptop and not long enough outside or doing other stuff so that's the next issue to address!
I was at where you're at twelve years ago so unlike some on here I still remember - it's not
that long ago! Rest assured that you should still get
a lot fitter and therefore should get faster. Either that or you'll be able to go further than your commute (maybe before too long a lot further than you'd think).
Or you might even just find that your speed doesn't improve all that much (I don't know - I was never timing myself and still don't), it just hurts far less or not even at all. Or to put it the way Eddy Merckx (I think - someone quoted it on here recently) put it, it doesn't hurt any less, you just go faster!
I cycle for transport and my 'commute' - from the village to the town to the shops and back - is about the same distance as yours. Given that it's so short, once I've warmed up (although we probably don't get anything like warmed up over such a short distance) I go pretty much flat out if I can be bothered, and although my speed from week to week probably isn't all that different, there are times when I'm 'feeling it' in my legs and times when I'm not.
I've done far, far longer runs - I usually cycle to and from club rides while most of the rest take their bikes on their cars so sometimes a club ride for me is up to twice as long as the others ride - often closer to 100 miles than 50 - and once you've extended your distance beyond what you do on your commute, you'll likely feel it isn't as big a deal as it feels now (to put it mildly) and might just have the ability and feel the confidence to push harder and go faster.
Be reassured though that when I started riding with my local group I'd just been riding up and down to the town for a few years, for transport, like you've been doing. It wasn't long before I was going far longer runs, and that was only because I'd laid the foundation by doing short frequent rides for a while.
Sorry if none of that is very specific or helpful, I've just tried to be encouraging, but it probably all boils down to something Merckx or one of the other legendary racing cyclists (can't remember) said when asked how you get quicker: "Three words: 'ride', 'the' and 'bike'."
