This isn't hard...as Renovator said, put the sash in the frame, hold the hinge up to the sash and frame where you want it to be (on the outside of both)...then mark where the screw holes are. Next, remove the sash from the window, and transfer the locations of the screw holes to the bottom of the sash using a square. How far in the screw holes are, you determine by placing that leaf of the hinge on the bottom of the sash, lining up the holes with the marks you made earlier, and using a pencil to mark the new holes. If the hinge is mortised into the sash, trace around the outside of it, and remove that wood with a chisel to a depth of the hinge's thickness. Unless I'm totally off, when the hinge was originally in place, the only visible part from the basement side was the lower leaf on the window frame...so when the window was closed, the hinge had an L-shape profile. My own basement windows are square, and all hinged on the side, but the hinges are totally concealed as they would be on a cabinet with inset doors. Personally, I'd do the hinges from the top or side instead. The most simple solution for your original question is to just surface mount the hinge on both halves rather than just the bottom.

I would add: This list presumes you checked the structural elements of the room before the kitchen went in....but make sure they all remained in good working order following the kitchen installation...windows open and close, doors open and shut (back stops installed), light switches still turn the lights turn on and off, all molding still properly installed and intact. On the cabinet doors I would add to check that they open and close properly and stay fully closed. On the countertops, I would add to check for gashes, dings, cracks and chips and to particularly check where the countertop meets the sink. If intended to be "child proof" check that all locks etc are installed and working. Floor molding properly installed? Where is your fire extinguisher? Have you added its expiration date to your calendar? If installing a range make sure it was anchored so that it cannot flip forward. Are all appliances properly leveled? Save all of your manuals, warranty and any required proof of purchase into a location where you can find them, such as hole punched and put in a three ring binder...look at the warranties to make sure that there is not something unexpected in them that you should be aware of....like required cleaning or maintenance. If there is any, put it on your calendar NOW. Also put in the name and phone numbers of the subs so you know who to call.

Widen the steps, create a wide landing, and widen the walk. Remove the small tree from smack in front of the window. Reshape the bed line. Move the tree (that sits off the right corner of house) out farther from the house and enlarge the bed below it. Create a circular bed of perennial below the tree. Have shrubs only below the windows (that don't extend beyond the window width.) Place color near the entrance. Fill the rest of bed with low groundcover. There is probably an opportunity for adding a street tree or two but this is not shown.

Taking it down and reinstalling is not a major delay. It will only get more difficult after the surrounding trim pieces are installed (but still not that big of a deal).