Realistically you are talking £1.5K for this trip. I had a few days at the end of the trip before the flight home, these costs have not been included.

Getting there and back

Return flights Heathrow - Punta Arenas - London City Airport - TAM Airlines £968.99
Train tickets to Heathrow from home £71.00
Train tickets from King Cross to home £59.50
Hotel in Punta Arenas (booked from home, one half of actual cost shared) £53.00

Spends in the UK

Freeze dried food taken to help keep food weight down £34.50

Spends outside the UK

Meal Santiago airport £14.00 
Transfer from Punta Arenas airport to Hotel$3,000 
Bus Punta Arenas to Puerto Natales$5,000 
Food bought at Punta Arenas$28,000 
750ml Methylated spirits / denatured alcohol$3,000 
Bus from Puerto Natales to the National Park (CONAF Administration)$6,500 
Park entry fee$18,000 
Camping at Cuernos$8,000 
Camping at Seron$4,000 
Camping at Dickson$4,000 
Dinner and half bottle wine at Dickson$4,000 
Camping at Los Perros$4,000 
Meal at Refugio Grey$10,500 
Bus to Puerto Natales$6,500 
Accommodation Puerto Natales (share of cost between two people)$19,000 
Evening meal - Puerto Natales$15,000 
Bus to Punta Arenas Airport$5,000 

Specialist Gear

Maps - three maps £45.00

Grand Total

(Chilean Peso {$} converted to £ at 6,500) £1482.99