Nothing wrong with riven slate in a kitchen or any other floor for that matter if its laid well like
this I assume you are using the two surfaces on the same floor?
Sealing will depend on the look you want. If you want to keep the slate looking like it is out of the box/crate then something like Lithofin MN Stainstop will do the job. If you want it to have more of a sheen and enhance the colour (wet the slate with water to get an idea of the look) then something like Lithofin MN Slateseal will work well. Don't be tempted to buy cheap sealers like plasplug or even Lidls
http://www.extensive.co.uk/section.php?xSec=18Apply sealer after laying then again after grouting.
Also remove any loose layers from the slate before laying and grade them for thickness if possible, a wide joint helps prevent oneven edges.
Jason