Quite possible fiction-factory, and your solution does work. I have to say however that the Daily Mail mindbender is rarely so complicated. I think it more likely that the answer of £50 given is incorrect. In my opinion, the most likely of the possible answers to be correct using algebra and trial and improvement is as follows:
L = £20
T = £60
M = £75
E = £150
K = £60
Total = £365
Mean = £73
This of course, is only one of many possible answers. If the question setter had included either, a comparison between K's rent and the rent of one of the others, or two different comparisons between any of the others and one other person, then a definitive solution could be found using algebra alone.
I have come across errors in the Daily Mail Mindbender before and was sad enough to email them once. I received a very nice reply confirming that there was indeed an error. No one is infallible I suppose.