LSE's official estimates:
Tuition Fees: $24.222.72
Room: $16,600.00 (this is 250 GBP per week for room and bills for 40 weeks)
Board: $4,980.00
Books: $1,992.00
Travel: $2,324.00
Personal: $5,312.00
Other Essential Costs: $4,980.00
Total Cost of Attendance: $60,410.72
At the current exchange rate, this is actually $56,407.60, but your loan is still for $60k plus, unless you ask for less.
They do say "maximum" costs for the cost of living, the two flights are 1,500, and that's 1,000 for a computer and 500 for health insurance. I don't know why you'd need the 500 for health insurance unless your parents aren't insured and you're going back to the US for holidays, as from what I'm aware NHS is free even for overseas students.
I highly doubt there's any way a student could spend $60k on the cost of living, unless that included jetting off to the Continent every weekend, but I'd still get the maximum amount of loan just in case there were unexpected circumstances.