De-gloss the existing vinyl floor by going over it with your vibrating sander and medium sandpaper, until it doesn't brightly reflect light. Thoroughly mop the floor with a household cleaner. Let it dry.
Divide the floor into four even square sections by snapping two intersecting lines across it with your chalk snap line. Square the lines to each other at the intersection with a t-square as you snap the second line.
Spread flooring adhesive over a few square feet where the two lines intersect in the middle of the floor. Set four vinyl tiles in place at the intersection. (You should still be able to see the chalk lines through the mortar.)
Spread additional mortar and lay the rest of the tiles, using the lines as guides to keep the grid straight and building out toward the walls. Lay each tile by setting down one edge first, butted against the adjacent tile, then lowering the rest of the tile into place, so you're not squeezing adhesive up between the tiles.
Lay all the full tiles that will fit in the room. Let them set overnight. Install the final tiles around the edges of the room, measure each space and cut the tiles on your tile cutter to fit.