Solution:

Fix the colouring of the cells and the starting position. We prove that Jana can reach any cell. Call a series of moves legal, if she starts from the starting cell with a teleport move, and uses the two types of moves alternately. Divide the cells into four categories.

- Call a cell teleport reachable, if Jana can make a legal series of moves finishing in this cell, but all such legal movement ends with a teleport move.
- Call a cell adjacent reachable, if Jana can make a legal series of moves finishing in this cell, but all such legal movement ends with an adjacent move.
- Call a cell easily reachable if it can be reached legally such that the last move is a teleportation, and also it can be reached with the last move being an adjacent move.
- Finally, call a cell unreachable, if Jana cannot move to this cell with a legal movement.

For consistency, we assume the cell that Jana starts on to be reachable with an adjacent move that happened before the start of the game and forces Jana to perform a teleport move next. The starting cell therefore is either adjacent reachable or easily reachable. Also, for any given cell, we call the other cell with the same colour its "partner cell".

Lemma. The number of teleport reachable cells and adjacent reachable cells must be the same.

Proof. Consider a cell $T$ and its partner cell $T'$. If $T$ is easily reachable, so is $T'$ (as any move before teleporting from $T'$ to $T$ must be an adjacent move to $T'$, and any adjacent move to $T$ can be extended by a teleport to $T'$). Similarly, if $T$ is teleport reachable, $T'$ must be adjacent reachable and vice versa. If $T$ is unreachable, so is $T'$, otherwise it would contradict the previous observations. Each pair therefore contributes equally many adjacent and teleport reachable cells.

Lemma. Each neighbour $N$ of a teleport reachable cell $T$ must be adjacent reachable.

Proof. A legal movement to $T$ can be extended with an adjacent move to $N$. But if $N$ was easily reachable, we could extend a teleport movement to $N$ with an adjacent move to $T$, so $T$ would be easily reachable too.

Lemma. Neighbours of easily reachable cells are never unreachable.

Proof. At the very least we could extend a teleport movement to the easily reachable cell with an adjacent move.

Assume that there exist one or more unreachable cells. Then somewhere on the board, there are two adjacent cells of which one is unreachable and the other is not. By the second and third lemma, this neighbour cell can neither be easily reachable nor teleport reachable, so it must be adjacent reachable.

Put a domino covering these two cells, and tile the whole table with dominoes containing this domino. Clearly, we can do this: Extend the two short sides of the domino such that the sides of the domino now partition the board into four rectangular regions. Since each of those rectangles have at least one even dimension, we can cover them by dominoes.

Now by the second lemma, every domino that contains a teleport reachable cell must contain an adjacent reachable cell, and there is a domino that contains an adjacent reachable cell with no teleport reachable cell. This is a contradiction, as the number of teleport reachable cells and adjacent reachable cells are the same by the first lemma.