athmō > aþumo is attested in LVS, but it’s unclear to me where the u comes from: perhaps merely a connecting vowel to separate the sounds. khithme > hiswe/hiþwe is attested in the Etymologies and -wa seems frequent in later nouns from -ma, so *haþmā > *haþwa seems good to me. The gloss “(lit.) cure-object” also makes sense to me.

+Robert Reynolds I take aþumo to be from *aþamo < ATHA the same way sanguma is (I think) from *sangama but I cannot prove this.

I agree. *haswa is better than *aswa since HATH implies something that helps to cure...ATHA is to heal.

*haswa looks good to me. It fits the root meaning pretty well. The only other thing I can think of that it might mean is “medical instrument” but that seems like a worse fit to me than “medicine” given the likely nature of Elvish medical practice.

I think "medical instrument" as a term is too specific to express it in a single word, I find it unlikely that a language would have a term for it. Haþalea tamma for medical tool, haþalea sorasta for medical equipment would be sufficient, and specific tools like scalpel or forceps would have specific names.