I got a 2:1 in Economics from LSE (2002) and took the MSc Economic and Social History at Oxford (2003-04) so I definitely think you have a strong chance, particularly if it's a good 2:1.
It mainly depends on the relative strength of applicants in your year but I know that the academics here have a lot of respect for LSE degrees and admit lots of LSE people (there were two other LSE students admitted in my MSc year of roughly twenty students although one of them later switched to economics).
As for the amount of Economic History in the course, that is up to you and the options you choose. The compulsory courses are more 'Social Science History' courses and are broader-based than what you get at LSE (which I found very interesting) but you can specialise a lot in 'pure' economic history for your dissertation and two optional taught courses. I probably did about 80% economic history and 20% general social science history in the course but most people take a more broad social science-y approach.