In my opinion, applying for ILR would be a waste of almost a thousand pounds. You clearly had a break in lawful residency.
Continuous residency is defined as:
• was absent from the UK for six months or less at any one time, and
• had existing leave to enter or remain when they left and when they returned
Breaking it is defined as:
• been absent from the UK for a period of more than six months at any one time, or is absent from the UK for a shorter period but does not have valid leave to enter the UK on their return, or valid leave to remain on their departure from the UK
Your LTR expired 12 July 2010 so when you left the country 8 days later you did so without existing LTR. That part is key:
You left the country without valid leave to remain. There is a 10 day discretionary grace of 10 days break if you received new LTR by the 22nd of July whilst in the country, but you didn't. So even if your absence until January were considered to be under 6 months, it began with an exit that was without valid LTR. Your new leave was granted in May 2011 and it seems to me due to the rules that your clock restarted then.