Hi Bill: It sounds like the water collecting in the corner of your garage is coming up from underneath the slab, because it can't drain under the footing on that side of the garage and run-off down the hill to the alley....yes?
If so, the way I would tackle it is as follows. Mark your floor 18" out from the wall with a heavy pencil. Rent an electric jackhammer (the Hitachi I have found to work the best) and break up the slab back to your pencil line. I would do this several feet on each side of the low corner, or until you get to dry soil/sand under the slab. Then dig down along the foundation about 12" or so and install 4" slotted plastic drain pipe, sloping it down to the low corner. At that corner dig a hole deep enough to sink down one of the fiberglas sumps available at Home Depot, and run the slotted pipe along the foundation wall and into the sump. Cover the pipe with builders mesh, cover that with p or 3/4 minus gravel, tamp down well. Take the gravel line up to the bottom of the existing slab. The top of the sump should be at the level of the finished slab. Then mix up concrete, fill and finish the rest of the trench. Put a small submersible pump into the sump, with a PVC exhaust pipe from the pump up the sump and out the wall.