This is the problem, £20 will get you some of the way, but to really get the results you need, you require machinery.  Pop into YouTube - polishing Aluminium and you'll see why machinery is needed. 
  
The initial process is by hand, but to get a top notch finish, our little blessed hands don't cut the mustard. 
  
By all means, have a bash, at least I respect you for giving it a go.  Don't expect to get the results you want though over a weekend.  Also as mentioned by AliveBoy, the pitting is the nightmare.  These need to be removed to get a perfect finish.  Then, to top the lot, you've got the awkward areas, which are truly a bastardo.   
  
Give it a bash and see how you fair. 
  
Happy to help with tips.