LoungeBuddy quotes $45 for the TK lounge, $59 for LH. TK is also closer to your departing gate and has a better view, so this should be an easy call. Be advised that you may have to wait for a bartender (LH is better in that apect) and that the food is consistently great (don't forget to grab some Turkish Delight at the front desk, especially if you've ever read The Lion, the Witch, and the Wardrobe). There's also a shower there, and if you want to nap the lounges in the quiet nook by the bar area ought to help.
Your timing also ensures you'll avoid the worst of the early-evening crowds. By the time you arrive, most of the evening bank of TATL flights will be well into boarding.