I would find the factor of 2 to be normal as it is cancelled out by the 1/2. The purpose of 1/2 is to eliminate the double counting. This is typical for pair interaction potentials.
If you're comparing with the book, there is a possibility of a typo.
Best to express the term in brackets (where the sigmas and R's are in) as something like U_ij and then manually perform a double sum and collect like terms. You should see the N pop out. Note that for this to work U_ij = U_ji as you stated.
May I suggest you use the chain rule of derivatives since V=1/rho=(rho)^-1. Take the derivative of the outside function times the derivative of the inside.