In that video, he used the T permutation (perm) to switch the two edges in the middle 2x2x2 block. On the 2x2x2 cube, the T perm switches the two corners in the upper right side of the cube. On the 2x2x4, those corners are edges and the algorithm still switches them. The T perm is R U R' U' R' F R2 U' R' U' R U R' F'.
In the video, he also mentioned the Y perm. It is F R U' R' U' R U R' F' R U R' U' R' F R F'. It switches the back/left and front/right corners in the top layer on a 2x2x2. I'm guessing that if you do that on the 2x2x4, it won't change the shape in the end like the T perm did.
There are shorter T perm and Y perm alternatives for the 2x2x2, but I'm not sure if they would mess up the shape of the 2x2x4 if you used them. If you want to try them, here are two of them.
T perm - U' R' U L' U2 R U' R' U2 R L
Y perm - R U' R' U' F2 U' R U R' D R2