Cleaning up dispersed potassium hydroxide (KOH) spill?
You just encountered a situation where you did something without sufficient protective gear; in this example, if you had known the battery contains corrosive material, you would have used protective gloves (for your hands) and a pair of safety glasses (for your eyes).

Your estimate that the Leclanché and alkaline battery contains lye is correct. The sensation of the skin turning slippery is expectable, since $\ce{KOH}$ started to degrade the protective coating of your skin (fat) into soap.  Rinsing promptly with plenty of water was a good action.

The lye equally attacked your cloths (one reason chemists wear lab coats), and as your skin, it is beneficial to rinse them right after the skin with plenty of water, too; and then to wash in a washing machine.  On contrast, bringing them as they are into the washing machine only will spread unnecessarily the contamination to other clothes.  Expect that the areas touched by the lye eventually will become brittle, especially if i) they "dried" for some time prior to rising or washing and ii) upon action of heat (e.g. ironing).

Regarding your study, I recommend to go the extra mile, and and clean all the possible surfaces; albeit hosing sounds a bit overkill (esp. inside an appartment).  You want to eliminate the pollution quickly, as it may attack wood and wood stain; and imagine you accidentally glide a book over such a forgotten droplet, it will be attacked just as your skin was attacked, too.  (Even more, as in contrast to human skin, the dead book does not have the ability to re-cover itself by the thin film of protective glycerides.)  In addition, droplets of lye do not dry away like water, $\ce{KOH}$ is hygroscopic and will constantly remain somewhat wet (and corrosive).