It's a shame you didn't ask this question before buying the house, OP. If you'd spent £60k more on a house that didn't need work done, you could claim UC straight away.
It's a notoriously difficult area to give advice on, because there are no hard and fast rules and a lot depends on the individual decision maker.
Having a house rewired because the electrics are dangerously old would be fine imo. If the existing kitchen is functional, but just old, the DWP may regard that as intentional deprivation of capital. And I wouldn't like to give a view about money spent redecorating and making good following the rewiring.
I dealt with a case a few years ago where a client's UC had been stopped because they suspected intentional deprivation of capital because of spending on home improvements.
Luckily, the client still had a copy of the estate agent's particulars that showed that the house was pretty much a wreck when it went on the market. The DWP were happy to disregard the spending on installing a bathroom, rewiring, central heating, new windows and doors, refitting the kitchen and the subsequent redecoration required. The client had kept all the quotes for each of the jobs, so we were able to show that the amount spent was "reasonable", ie not the highest estimate, and all the receipts that proved that that was what the money had been spent on. Where a property has a functional bathroom, kitchen etc, I suspect that the outcome might be different.
But the DWP aren't bastards in this respect, I had another case where they were happy to consider a couple of grand spent on a holiday in Spain reasonable. I guess it would have been different if it had been a couple of weeks in the Maldives. In another case, they disregarded £10k spent on a second-hand car.
Once the money you have is below £16k, you will probably still have some UC entitlement. To check, go on to entitledto.co.uk and put in all your circumstances and savings of £15,999. That will show what you would get once you're below the £16k cut off point, and the amount of UC will increase as your savings drop further (as long as you remember to tell them and provide evidence!).
Hope this helps!