That's a sealed cartridge bottom bracket. The bad news is that if there's play in the spindle, it's shot. The good news is that they're inexpensive and easy to replace. A Shimano UN55 replacement for that should cost you in the neighborhood of $25. Measure the length of your current spindle and find a replacement that's sized to match. (Assuming your current one's spindle is correctly sized.)
To remove and replace, you'll just need a wrench and a bottom bracket tool (the "spliney tool" you mentioned) like a Park Tool BBT-22 or BBT-32.