I mean, how difficult is it to keep a hospital clean?

Honourable Fiend has the answer:

About 12 years back, Fiend spent some time working at Addenbrooke's Hospital in Cambridge. During that particular Tory-era NHS experience one particular ward had a recurring problem with MRSA infections.

So they emptied the affected ward, and the one on each side. All three were denuded of everything. The beds were disposed of. The laundry, incinerated. Every last piece of furniture removed and destroyed. Affected and potentially-affected patients internally quarantined for pre-emptive treatment with antibiotics.

The three bare-shell rooms remaining were steam-washed, hosed down with disinfectants from corner to corner and re-floored and repainted from scratch. Beds and other equipment were replaced with brand new stock.

And after all this? The next case of MRSA took four days to materialise.