The border between Togo and Benin can easily be crossed from Kara. The road is good and can be covered by taxi. You can book a taxi-brousse all the way to Djougou, though you may need to change vehicles before the border. Alternatively, take a taxi-brousse to the border, walk across (about 5–10 minutes), and pick up onward transport on the Beninese side.
The crossing itself is straightforward, but you must arrange your Benin visa in advance — it is not possible to get one at the border.