You have a constant vector field over a flat surface so an integral is not needed. You area vector is given by a unit normal to your surface multiplied by it's area. You found the the a normal vector to be <2,0,1> if you normalize this you get

by the luck of the draw the area of the rectangle is also so the Area vector

The flux of a constant vector field through a flat surface is the dot product of the area vector and the V.F so we get

IF you really want to use an integrate the (from your equation of the normal vector) the plane has equation 2x+z=2 or z=2-2x

Then use the formula Flux =

Where dA is the projection of the surface into it's domain so in this case the xy plane as you wanted.