===== Answer =====

He is losing $2.60

To start, you need to know how wrong the clock is.  Begin by calculating how long it takes an accurate clock's hands to cross.

Hour hand:   360<sup>o</sup> / (12hr * 3600sec/hr) = 1/120 <sup>o</sup>/sec \\
Minute hand: 360<sup>o</sup> / (1hr * 3600sec/hr) = 1/10 <sup>o</sup>/sec \\
relative speed between the two: 1/10 - 1/120 = 11/120 <sup>o</sup>/sec \\
time per overlap: 360 <sup>o</sup> / relative speed = 360 <sup>o</sup> / 11/120 * 3600sec/hr = 12/11 Hours \\

If the broken clock makes 1 overlap per 69 minutes, and the good clock makes 1 overlap per 12/11 hours, we can get a 'time dilation factor' of 69/(12/11*60) \\
This leaves us with a simple equation to find how much money he is losing or gaingng: $6 * 8hrs * 69/(12/11*60) - $6 * 8hrs = $2.60. Since the broken clock moves slower than it should, he is working more than 8 hours, so he is losing $2.60.