the Eurofighter is simply not designed to do that and would, most likely, crash attempting such a manoeuvre. Remember that it doesn't have to just touch it's wheels down for a short time: it would have to remain in contact with the ground for a whole mile. The slightest turbulence and it would either loose contact with the ground or crash into it.
1200mph is just over 500 meters per second. The flying KM (not a mile) that the record is measured over would take under two seconds.
Flying just above the ground is one of the most stable flight modes for aircraft - the ground-effect forms a cushion of air that the aeroplane can cruise on. At 1200mph, you need pretty flat ground, which is exactly what we have here.
Edit to add: I'm pretty sure the vehicle has to be 100% piloted/driven by the person on board.
That does make a big difference, and would clearly exclude most modern aircraft. Are you 100% sure Bloodhound and Thrust SSC have a strictly mechanical linkage between the steering wheels and the driver?
Thrust SSC had two jet engines, one on either side. Undoubtedly it had electronic / computer assisted thrust balancing, and I'd call that quite a big help in steering the car. Remember, 'steering' here means just going in a straight line. No turns or corners needed.
You do some somewhat determined to degrade the project or make it seem trivial. Why?
I admire Bloodhound, and I loved Thrust SSC, I just don't see it as a car, more of a low-flying rocket with wheels or an aeroplane with fixed undercarriage. That doesn't make the people who design, build and fly them any less than 100% heroes.
Bloodhound and Thrust SSC are far more dangerous than aeroplanes - when I travel at 400+mph, I'm always a safe mile+ away from the ground and any other hard objects. Not dangling an inch from the hard ground like these insane folks.
But to me, a LAND SPEED record means propulsion via pushing against the ground, full stop. That means a high speed train, running on rails at 1000+ mph would be fine for a land speed record, as long as the propulsion power was through the wheels. But the FIA would probably disallow it because they're car nuts. I seem to recall they specifically say rails are not allowed. I'm more open minded than the FIA here
The meaning of propulsion from reaction against the ground might also include an electro-magnetic rail-gun style land speed attempt.
Propulsion is from the EM field, with the vehicle pushing against the field exerted by the rails, which form part of the ground. To be fair, you could say that all the energy for the EM fields must be generated on board the vehicle, and not supplied by external generators, which would make a rail-gun (or coil-gun) scenario a much, much harder idea.