Hi there Jat, hope you are well!!
It may be a little difficult getting a course for just 1 week. a lot of schools demand a minimum 2 week stay. However if you go to Madrid there are scores of schools to choose from, many at very good rates. The best idea, as soon as you get to Madrid (after checking in your hotel) is to go to an Irish pub and pick up the free English-language publications - they will have loads of schools advertising.
As for 10 days being too much in Madrid - Believe me 6 months is not enough. If you do want to escape the big city there are scores of destinations you can visit on a day trip (Avila, Toledo, CHinchon, Segovia etc). Simply do a search on day trips from Madrid on this board.
By the way, you're not from Leicester are you (it's just I knew a Jat once who loved visiting Spain)
Good luck in whatever you choose....
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An English Bookseller in Madrid