lose the syringe... put the reservoir back on, have it sit at a spot you can get at it. fill it up to the top. Put a piece of hose on the bleeder nut. With the bleeder nut closed, pump up the brakes. The crack the nut as you pump the lever down. at the bottom of the stroke, tighten the nut, repeat and repeat and repeat, making sure the reservoir doesn't empty. Eventually you will see the air bubble and new fluid come through, tighten the bleeder nut and then pump up the brakes again, should get solid. On the first ride, may need tp pump them up one more time as the pads re-bed in.
Edit... you can use the syringe on the bleeder nut to provide a bit of vacuum, when you pump the brake lever, it will draw fluid.