I know Corgis are notorious about having ticklish feet, and I know that I have related my problems with getting Jack to let me do his nails.  

Here's my blog post detailing the method I used.  SImilar to clicker training, but I used a marker word instead of the clicker to reward him for the tiniest steps towards letting me do his nails with a dremel.

http://www.mycorgi.com/profiles/blogs/jack-let-me-dremmel-his-back

The back feet were pretty easy, but I've been working on the front for 5 months now.  The biggest thing that had to change was my mindset:  I got over the idea that he should just let me do this and turned it into a little way to spend five or ten minutes working with my dog a few times a week.  I decided I didn't care how long it might take, and would not mind if we spent weeks at a time with no improvement: I was just going to sit down once or twice a week and do this.  

Except for three-week break when I was sick recently, I spend at least one weekend day and frequently one or two weekdays working on this.  We were mostly working on the left front, but after hitting a wall I switched up both the treating method and the foot.  He seemed relieved when I switched to his right front.   And I started using string cheese as the treat, and giving lots of jackpot rewards (five or seven treats in a row) for ANY improvement, no matter how slight.

And this evening, I was able to do an entire front paw.  Not just touch with the dremel, but actually grind them down to a reasonable length.  One toe seems particularly ticklish (and I got my own thumbnail while trying to hold his steady).  

I was SO proud of my boy, and it's so nice to look at a neat paw where the foot can hit the ground evenly without a too-long nail.   In the summer the pavement keeps them short, but in the winter he walks so much on snow and stuff that they get too long.

I can't promise the method will work with every dog, but Jack is terribly claustrophobic and when we started this process he would rear up and jerk his paw away if I even tried to hold his foot.  This is not a dog you could ever hold still by force because he panics.  

So lots of patience, marker-word, lots of high-value treats, and most importantly my deciding that we had absolutely no timetable to get this done worked for us.  I'm hoping maybe my experience will help someone else whose Corgi is difficult about nails.