Without seeing the spreadsheet, I can't tell you the exact formula to use.

However, under 'Data' there is a 'What-If Analysis' menu. Selecting 'Goal Seek' lets to choose which cell is the target cell, what the value you want of that cell to be, and which cell to change to reach that goal.

So by designing the sheet to figure out the list of initial costs and monthly costs, then have the monthly costs multiplied by a 'month' cell, you can set the spreadsheet up in such a way that Goal Seek will change the number of months until the difference in price is $0.

Once you get that part working, you can record the actions as a macro and have a command button to run it so the client doesn't have to do any manual operations at all.

Edit: See my attached example. I have 4 columns, one 'Recurring Costs' and one 'Non-Recurring' for both the current and proposed setups. Plug the costs in. Sum the columns. Subtract the proposed non-recurring from the current non-recurring to create the non-recurring difference. Subtract the proposed recurring from the current recurring to create the monthly difference. Create a cell for the number of months required to recover costs and put a fake value in there. Create a cell for difference in price using the formula 'Non-Recurring Difference + (months * Monthly Difference)'.

To do the optimization go Data -> What-If Analysis -> Goal Seek and the Set cell is the Difference in price (B14), To value is 0, By changing cell is the months (B12).