Bone is an interjection used to express a nonenthusiastic aggreement, like English okay, Polish dobrze and Russian хорошо – all four of these (including okay) are also adverbs, which literally mean well in the literal sense.
Well used as an interjection in spoken English has a different role, it introduces a minor disaggreement or something contrary to expectations, similar to Polish cóż, Russian ну or Esperanto nu.
I can't give you a grammatical explanation but I've heard it quite a bit.
German also uses "Gut" in this sense: "Gut, ich werde es selber sauber machen". (French, on the other hand, uses the adjective form, I believe: "Bon, je le ferai moi-même."
Perhaps a good reason would be that Esperanto adjectives are used only in relation to nouns but adverbs in relation to phrases and sentences (e.g. "Kuri estas bone", to run is good, where "kuri" is not a noun), and here it applies to the situation as a whole rather than to a particular noun.
This smacks of English (or as mizinamo points out) German sneaking into Esperanto to me also. I could see a non-English Esperantist reading this as I will clean it well myself. As in "I will do a good job of cleaning it, myself." And with the Bone put first to put extra emphasis on the fact that I am, myself, going to do a particularly good job of cleaning it.
If "fine" and "okay" are acceptable as English translations of "bone", why is "all right" rejected?
Several reasons. "Ok" in Esperanto means eight, and "aj" is the ending for a plural adjective. And adverb is needed here (see Vylah's comment). So it would have to be "oke", but that Esperanto word already means "eighthly".
...and anyway, "oko" doesn't really sound enough like "O.K." to make it recognisable to people who know "O.K." from English.
I can actually conceive of a situation in which the plural of "that" could be used (I don't yet know enough Esperanto to completely translate this, so sorry for English here):
Person1: "Well, at least there's only one of that."
multiple thats start arising dramatically from the ground
Person2: "No...I think there are a few more thats than that."
While unlikely, it's theoretically possible to find it in a sentence.
I think you'd have to explicitly turn it into a noun to do that in Esperanto, though: keoj or ke-oj or something like that.
I don't think -ej can occur as a suffix in Esperanto at all, much like -un or -es. (Outside of correlatives which have their own system of endings and where words such as kiun and ties do, of course exist. But you can't tack those endings onto other stems.)
It would be incorrect, because the Esperanto word for "to clean" is "purigi". "Lavi" is a specific sort of cleaning - washing. (You don't was leather shoes, wooden furniture, or woollen carpets to clean them, for instance).