This is taken from the LSE website:
Year 2 Entry
Entry to Year 2 of a LSE course is extremely rare, except in exceptional circumstances. In most cases, the School is oversubscribed for its courses and cannot accommodate extra students.
Departments who do not accept applicants into the second year include:
LLB Law (M100)
Government (L230, LL12, LV21)
Statistics (Actuarial Science N321, Business, Maths and Statistics G0N0)
However, some departments are willing to consider applicants on an individual basis, if they meet the stated entry requirements. Applications for entry to Year 2 should be made in the usual way through UCAS.