Replacing sill (bottom plate) of wall
You need to set up a temporary wall under the ends of the ceiling joists of the garage, bearing onto the floor of the garage. Since the section needed is so short, you will only need 5 or 6 2X4s or 2X6s to hold the ceiling and lift the wall for the short time while you replace the bottom plate.
 If you need 10 ft studs to hold everything up, cut 2 of them in half to double up as top and bottom plates, then use the other 3 to use as studs cut 1/4 longer than needed for a snug fit to lift the existing a little. Get one stud jammed in, set another and with either a hydraulic jack, or by "bowing" them in, as in pushing in the middle of the stud to bow it so it in essence gets shorter between the 2 points until you release it, which will lift the wall a little. work all the studs together one at a time, one will help the other.
Or 2 bottle jacks can do it all too, if you have 2 handy. 7 ton, maybe even 2 1/2 ton would work too. That corner, if there is no more than a roof over the garage, is pretty easy to lift up. You wont need much, an 1/8" will do. 

This is all on the premise that the loft is contained under the roof with no 2 story bump out to house it, that the roof rafters bear on the top plate of the same wall that has the destroyed bottom plate.