Who told you it would be £80?
It would only be £80ish if you want an anytime single, which you'd only need if you don't know which train you're going to get and you want to travel at at peak times.
The Cheapest advance single currently available to London is £24, on the 10:00. You can buy this from
East Midlands Trains or any other train company. Don't use Thetrainline or it will cost you more, as it always does with them.
If you want tube travel to Heathrow included, the cheapest tickets are £26.60 on trains that leave Leicester at 11:00 and 11:32
Or if you can get the 12:00 train there's a £15 ticket to St Pancras available from
Megabus
The above cheap tickets would need to be bought quickly as the price will go up once the cheaper ones have been sold.
If you can't commit to a specific train, but won't be travelling in at peak times, the cheapest ticket is £58, which is valid from the 10:24 train onwards. There's no need to buy this ticket in advance, you can get it at the station on the day.