Descend the steps towards the Cafeteria and summon Eleanor to your side. Take out the Splicers inside the room opposite the Vita-Chamber and hack the Security Camera on the right before going any further. Cross the Cafeteria and continue through the kitchen and down the stairs to the maintenance tunnel. Melt the ice near the freezers to collect a number of Proximity Mines, then turn your fireballs on the Brute Splicer across the room.
Test out your new Telekinesis 3 Plasmid with the Leadhead Splicer up ahead—grab her close, release the Fire Plasmid button halfway to move her in front of the reticule, then open fire! It doesn’t get any easier than that! Continue through the dripping Laundry Room to the stairs and chase the Alpha Series—Sinclair, presumably—up the steps.
Use Telekinesis to get the Auto-Hack Dart from above the pipes in the showers, then continue east into South Quad of the Cell Block. Sinclair is in the Security Tower on the left.