Killing 11 hours time in Heathrow, London Airport

I don't know about the documents. By far the quickest way to get to town is the Heathrow Express https://www.heathrowexpress.com/. Leaves at Terminal 1/2/3, 4, and 5 goes non-stop to Paddington station. You can plan journey on public transport here http://www.tfl.gov.uk/plan-a-journey/ . Trip from airport to most places in central London will be about an hour or thereabouts.

I'd recommend Underground and trains over cars and cabs since London traffic is a mess. I had once a car from Heathrow to Soho Sqaure and it took about twice as long as public transportation would have been.

It is not possible to know for certain whether you need a visa because we do not know where you are traveling or why. There are many circumstances under which you might be eligible for transit without a visa. If you do not want to pay the visa application fee just for a short trip to London, you might prefer to risk being accepted under these terms (from https://www.gov.uk/check-uk-visa/y/india/transit/yes):

Transiting without a visa

You might be eligible for ‘transit without visa’ if:

• you arrive and depart by air
• have a confirmed onward flight that leaves on the day you arrive or before midnight on the day after you arrive
• have the right documents for your destination (eg a visa for that country)

One of the following must also apply:

• you’re travelling to (or on part of a reasonable journey to) Australia, Canada, New Zealand or the USA and have a valid visa for that country
• you’re travelling from (or on part of a reasonable journey from) Australia, Canada, New Zealand or the USA and have a valid visa for that country
• you’re travelling from (or on part of a reasonable journey from) Australia, Canada, New Zealand or the USA and it’s less than 6 months since you last entered that country with a valid entry visa
• you have a residence permit issued by Australia or New Zealand you have a common format residence permit issued by an European Economic Area (EEA) country or Switzerland
• you have a residence permit issued by Canada issued after 28 June 2002
• you have a uniform format category D visa for entry to a country in the EEA or Switzerland
• you have an Irish biometric visa and an onward flight ticket to the Republic of Ireland
• you’re travelling from the Republic of Ireland and it’s less than 3 months since you were last given permission, on the basis of holding a valid Irish biometric visa, to land or be in Ireland
• you have a valid USA permanent residence card issued by the USA on or after 21 April 1998
• you have a valid USA I-551 Temporary Immigrant visa issued by the USA (a wet-ink stamp version will not be accepted)
• you have an expired USA I-551 Permanent Residence card issued by the USA on or after 21 April 1998, with a valid I-797 letter authorising extension
• you have a valid standalone US Immigration Form 155A/155B issued by the USA (attached to a sealed brown envelope)

You won’t be able to transit without a visa if a Border Force officer decides you don’t qualify under the immigration rules. You can apply for a transit visa before you travel if you’re unsure whether you qualify for transiting without a visa.