Skiing near Frankfurt 18-23 February

Have a look at this map: https://www.bergfex.de/deutschland/

As you can see there are two really small ski areas in the Taunus mountain range right next to Frankfurt. But the chances of them being open while you are there are essentially 0. Note that those small areas would also usually only be open on weekends.

So this leaves you with a bit of travel. If you have a car, you can reach the Hochsauerland area (Willingen, Winterberg, Neuastenberg. Elevation 600-900m) to the North, the northern Black Forest (Mehliskopf, elevation ~1000m) to the South or the Rhön mountains (Wasserkuppe, elevation 950m) to the East within 2 hours. They should all have snow around that time. However, with climate change you never know. Willingen at the time of writing is pretty green (https://www.skigebiet-willingen.de/page/webcam.php), but they seem to be using artificial snow, so you should be able to ski the biggest slopes.

The closest ski area to surely have natural snow and some more advanced slopes would indeed be the Feldberg area in the southern Black Forest, around 4 hours by car.

All those areas are kid friendly. No dangerous slopes and lots of ski schools around. I've been there.

You can find here the ski resorts in Hessen (is the State of Frankfurt), but i think there is in that time very little snow. You can go to Feldberg in the Black Forest, which is 3 hours from Frankfurt. I think there is a lot of snow and there are very good ski resorts.