Your UK Visa should cover you for England, Scotland, Wales and Northern Ireland
, but if you wish to visit the Republic of Ireland (not part of UK) which includes Dublin, Cork, Kerry etc., you will need to contact the Irish Government for their visa.
As far as Ireland is concerned, I would highly recommed Cobh (pronounced Cove), which is where the Titanic has its last port of call.
From there you can visit Cork and Blarney Castle. Truly a beautiful part of the world, with wonderful people.
They will talk the hind legs off a donkey and you will find that a half mile walk will take three hours with a diversionary but compulsory visit to the local pub, but you will leave with a warm feeling for the place and a tear in your eye.
Other personal favourites of mine, off the top of my head.....
Wales? Portmeirion, Snowdon park and railway, lots of coastal beauty.
Scotland? Edinburgh, Fort William, Highlands.
North East England? Lindisfarne (Holy Island), Durham Cathedral.
North West England? Lake District, obviously the cities of Liverpool and Manchester.
Other places.....working sort of downwards.....
York (ancient and very interesting)
Yorkshire Dales - stunningly beautiful.
Warwick (pronounced Worrick) great castle, plus more historicsl stuff.
East Anglia (Norfolk and Suffolk mainly) where most of the American bomber crews were stationed in ww2. Flat country, lots of thatched cottages.
Cambridge. Go for a punt down the river.
Saffron Walden, an archetypal English Market Town.
And that only brings you halfway down England, and only scratches the surface.
A more complete list will probably develop here over time, as others add their favourites.
As far as your idea of a Hotel inside the M25 is concerned, I think you would be able to find a lot better value place OUTSIDE the M25, as inside knocks the rates up considerably.
If London is a must for you, and I tend to agree with fresco about other places being just, if not more, English nowadays, then I would strongly suggest finding a good location that has efficient train links with London and maybe stay somewhere 30 or 40 miles out and commute in on days when you want to do the City tour(s) as necessary.
Another good point made was that we can pretty much travel from the top of England to the bottom in one day (at a push), and certainly two days taking it easy with a night in a hotel on the way.
Do your research, select your desired destinations and then mark on a map what would be the best centre of routes for you, that's my suggestion.
Back again later.....