Hi OP, really feel for you!
White/brown noise is your friend. I paid a lot of money with my first to a sleep consultant and this was the biggest thing we took from it - as little one moves from deep to light sleep every 45 mins they are seeking reassurance all is well to stay sleeping. The noise playing continually in the room all night (NOT fading out after 10 mins, stupid Ewan the sheep) helps with this a lot.
Another thing: (I may get flamed for this but) would you consider trying him with a bottle or 2 of formula during the night? Our HV recommended it for getting more hours of sleep at night and in my experience it works - my 2 year old has always been combi fed, so still gets plenty of boob, just chugs back a big bottle of formula to fill his tummy up at bedtime and again at 2am...
I prep the bottles with cooled boiled water, measure out the formula (Hipp organic brand which doesn't need to be sterilised) in one of these: www.amazon.co.uk/MAM-Milk-Powder-Dispenser-Blue/dp/B00PDT9NLU
then can just tip the powder into the room-temp water at 2am and shake, bottle ready in 5 seconds! It's so easy and helped us a lot.
Also co-sleeping, which PPs mentioned, IMHO is much kinder to the little one than sleep-training. My little one loves the comfort of me sleeping beside him and is far less fussy than the few nights we've attempted him in his own bed.
Batch-cooking is easy if you have a freezer - I used to do chunky veggie mash, fill basically a big ice cube tray with 8 more 'meals' of it and feed him one while freezing the rest. Do the same with all your meals and then most of the week you just have to de-frost or re-heat from frozen, instead of properly cook. WAAAAAY easier, and makes it less miserable when it goes in the bin/on the floor.
Also - massive hugs to you, it's really hard being a first time mum and you're doing it in while processing a huge break-up, as a solo parent - you deserve every credit and to feel really proud of how much you've achieved. Hats off to you! It's not easy but every stage passes - quickly, in hindsight! Wish you all the best.