It doesn't hurt you. Both are excellent.
I got into Durham this year too, but I also got into Cambridge. If I had firmed Durham, my job prospects would not suddenly worsen to the point where I would not be able to get the same jobs as I would've if I had firmed Cam, just like going to Durham over LSE is not going to worsen your job prospects. At this calibre, no employer, embassy, government or whatever, is going to care that you went to Durham over LSE.
I *definitely* would not recommend reapplying next year because there is no guarantee you would even get another offer from Durham because it is competitive and there is definitely no guarantee of an offer from LSE, for which the competition is insane.
Relax. You're going to an excellent university.