Yes, a 7.5 m2 or larger rig would really help in 13-15 mph, but at a steady 15 mph, your 6.5 should be enough to get you planing if you are rigging it for power.
What is your weight?
If you are 5'6" tall (hey, that's my height as well) and weigh =< 180 lbs. your 6.5 (rigged full) should be enough to drag you up onto a plane if you keep your board really flat (rail to rail) and move back on your board "progressively". You can move back at a rate that keeps your board accellerating. If you don't continue to move back you may find that the nose isn't getting up high enough to incline your boards planing surfaces correctly and your board will "hang" and not accellerate more until you move back slightly more.
Or, you could be moving back a little too quickly and the nose pops up and your planing surfaces are inclined too much for the board to plane effciently.
Try to tie the rate that you move back to the accelleration of the board.
It sounds like you aren't beach starting when you have "big" wind. Beach starting (other than getting things aligned) gets easier as the wind comes up. So does waterstarting, and with only 13 knots of wind, you really aren't up to a windspeed that would cause any problems. Practice "aligning" your board, holding onto only the boom (or front hand on the mast rear hand on the boom) and turning your board around 360 deg. in both directions. Soon you will begin to develop the "feel" for what you need to do with the rig to get your board to align perfectly right next to your back leg. Then just step on and off you go.
If the rig is twisting around and trying to pitch you over to the leeward side, you are putting too much force in it. Open the rig up so it just supports itself, then learn to steer the board by the mast foot. Slight downward backhand pressure turns your board upwind, slight pushing or raising the front hand drives the nose downwind. Pretty easy once you get the feel for the balance.
Hope this helps,