Not really nearby, but once stayed at an aparthotel style place south of there, but still walkable. The apartment small but clean and decent, the area slightly iffy (but this is London, so much of it feels that way to me).
If planning to go their Friday and/or Saturday, consider staying in the 'city'. There's loads of apartments there to service the financial hub, but those folk all head home by Friday (and many make the journey home on thursday now). That can result in some attractive prices, in a quiet location, and it's about a 40 minute (moderately pleasant) walk from Liverpool St station to borough market. Downsides would be that many of the coffee shops, restaurants etc. shut on weekends, so it can feel very quiet, plus if there's going to be engineering work on the tube, it'll happen here on weekends.
I suggest an apartment, because it really is super food grazing at Borough market, so you'll want the flexibility to eat the food back at your own place. There are also wine, cider and beer shops in the market, and some fancy soft drinks as well, be in Kombucha or freshly squeezed juice, or doubtless other stuff. There's a gelateria (3bis) just outside the market (not far from Brindisa). It's decent, i.e. good by UK standards, but I reckon just average or a bit below by Italian standards.
Many good stalls, and few 'meh' ones, but the prices are usually high. Sometimes that's justified, sometimes not. The huge range of artisan foods really is a strong draw to me.
It can be good for seasonal foods, with the large greengrocers (near the service road that splits the site - Stark's IIRC) having a solid wild mushroom selection (oft photographed - just push past the instagrammers!), and the stall that is keen on foraging (fitz) generally does so well. There are truffles in season at both these stalls, plus the truffle specialist Tartufaia. They gave solid advice on black truffles many years ago, pointing me away from emilia-romagnan to Essex

. The extra freshness of more local food meant the local ones were more fragrant.
Superb cheese selection across 3-4 specialists (including neals yard). I really like the alpine stall/shop.
Gastronomica good for Italian stuff. Keep an eye out here and at another stall for smoked (wet) mozzarella, but smoked burrata also superb. I prefer their more traditional fresh pasta to Pasta Tua, which I've found stuffed too much. Bianca Mora good for Parmesan, and if they still do the wax covered salami, that was always enjoyable.
Olive oil stall very good, and if you hit the arrival of the 1st press, you'll encounter some lovely green peppery olive oil. If not, their Incantro is very much in that style all year round.
Monmouth coffee good, but not worth waiting in the queue for, nor squeezing into the tight interior. Best when it's quieter.
Plenty of bread options
Best days to go? It's probably changed, and certainly changed post Covid. In days gone by Saturday and Friday were heaving, but Friday got quieter as many worked from home on Fridays. Thursdays used to be a nice balance of moderately busy and most (but not all) stalls were open. Wednesdays was noticeably quieter.
I hope you enjoy it. It can result in a big spend, but you'll return with some very tasty food.