I don't think it will finish at 39-40 degrees, if you are getting any substantial fermentation at all. That's too low even for a true lager yeast, which the Kolsch is not. If you are going to use the fridge, you will need to get a temperature controller. If you're electrically savvy, you can wire in a fairly cheap controller ($30-40). If you want to make it easy, look for the blue pre-wired Johnson digital controlers (A419). I bought mine for $75. Then you can control to within 2-3 degree total range.
My second Kolsch, which was all grain like yours just finished carbonating in the bottles. Very clean with a subtle wine like characteristic which is true to style. It's very tasty and I fermented at 60 on the nose. Brought it up to 68 for the last 3 days to let it finish up and clean up any diacetyl. Then lagered at 37 for 4 weeks. Yum!
Primary #1: Empty #2: Black Currant Wine
Secondary #1: Berry Rhubarb Wine #2: Empty
Kegged: Cascadian Dark Ale
Bottles: Spruced Winter Warmer, Surly Bender clone, Big 50 Barleywine, Framboise Lambic, Dark Belgian Strong, Kicked by a Moose Scotch Ale, Russian Imperial Stout, Sparkling Elderflower Wine, Barolo Wine, On Deck: Orange Pale Ale IV, Hidden Lake Nut Brown, 2H IPA III