I think the discussion is heading into the wrong direction.
People here are misinterpreting Life Insurance as Health Insurance.
As per existing laws both the things are different and one company cant give both. So LIC is not into the health business.
General Insurance companies are responsible for it.
Health Insurance business is markedly profitable business for companies so day by day we are seeing so many entrants in this field.
I have my family covered with New India Assurance Co., an old government stalwart and as my whole hospital and professional package managed by them i feel comfortable deeling with one person only.
Other notable players are
Iffco Tokyo General Insurance
ICICI Lombard general Insurance
Reliance (Iam sceptical about their low priced products as we never they will pull a rabbit out their hat)
Royal Sundaram (they are pretty straightforward and few good products too)
HDFC Standard (i have found them to be most professional of any prvate sector players in general)
SBI has also started their General Insurance venture iirc.
One more player metro residing players consider is Apollo - DKV but i am not sure about their reliability.
an excellent article regarding health insurance and mediclaim :
Health Insurance Plans – Life Insurers vs. Non-Life Insurers
a comparative chart for mediclaims of few companies
all above info from apnainsurance.com