Go into Autotrader and see what they're saying is the current value of your car with its full year's MOT.
Then try Webuyanycar, Carwow, etc and see what they say.
If it looks like the £2k is worth paying then think about it, but also think about getting a second opinion. Go on your local FB pages and ask for recommendations.
At some point down the line the car wil not be worth repairing. That's the reality of owning older cars. My previous car was a 55-plate Jag which I got at three years old (a bit young to be driving, I know...) from the auction in Edinburgh for peanuts because it already had 125k miles on it. I took it up to something like 275k miles over the next four years (35k miles pa just commuting to work) but it was costing more every year to get it through its MOT so I decided I had to move on. Its final MOT showed up rust under the sills and elsewhere underneath but otherwise ok mechanically so I got it seen to independently, got the resit, which it passed with no advisories, then traded it in for more than double what WBAC were offering (plus paid £1k less for my Mazda than any other garage was selling it for).
My wife's old Jazz (57-plate) was handed down to Orchidette when it was about eight years old with 80-oddk miles on it and going well but it started to cost after that and we eventually advised her to scrap it when its next MOT showed up stuff that was going to cost a lot to fix. We/She got £475 from a dismantler.
My 61-plate Mazda tends to get through MOTs with only wear-and-tear issues these days but I'm down to 5k miles pa and Mrs O only does £2k pa in her wee 63-plate Nissan Note. We've just taken out a lease deal on a Mazda 3 to replace that and will be moving that on via the approach above. It's been valued at £4500 and £3800, which strikes us as reasonable. It's a perfectly good wee car but not in the same league as the Mazda 3, which is why we're holding on to that for now. The Mazda is being valued at around £3500 but it's the reason we're sticking with the model for the new car.
The bottom line is that cars will always cost money. If I could live without one I would but that isn't an option where I live and I'm not the type to take to living in a biggish town.