God this sounds just like my neighbour. Her cat kept coming into my house so we eventually got in contact because of that. She is in her 70s and so a few times I went round to be polite - obviously lonely and a bit frail. Same as yours, quite rambly, couldn't get a word in edgeways and a cup of tea would take at least an hour and a half before I could make my excuses and leave.
Next thing I know, she's texting every day. Found me on Facebook, started tagging me in 5+ posts a day, sending me videos. She took everything as encouragement - I once took in a parcel for her, so she started getting parcels delivered to my home 'by accident' (so she could come round and talk more). It was bad enough until she started getting bizarrely rude. Comments on my weight (that I'm fat but seem to like it that way - and that my daughter seems the same), my health, my relationship. I told her I'd (amicably) split up with my partner and she'd constantly tell me I'd been 'brutally abandoned by my lover' and ask me if I was 'coping'. Whatever you told her didn't matter, she had her own delusions even on YOUR situations.
She ends up moving away, much to my relief, and leaving her cat with me. Three months of peaceful, blissful silence - and I wake up to paragraphs of text saying I'm cruel, horrible, that I've been deceitful and neglectful by not giving her cat updates. She even told me she'd knocked on my door multiple times (it's a 45 minute journey for her) and I hadn't answered. She never asked for them before leaving and had never text asking for them. Now I send her a picture every few weeks - on her request - and she takes it as an excuse to ask me really personal, invasive questions I don't want to answer. It stresses me out no end. Anything and everything I do she takes as encouragement. I'm scared she's going to ask to come round for a visit and she'll continue to push and push. I have no idea what to do anymore.
Genuinely, if anyone has any advice I'd be grateful...