Audio Equipment and Home Theater Audio : What jack is this? And how to replace it?

What jack is this? And how to replace it?

I bought a cheap hearing aid device from China for my elderly mother, and it actually works pretty good except that the ear bud falls out of her ear and it's hard for her to manage.

I would like to replace it with another headphone that is more stable on her ear or her head, but I don't know what to buy or what adaptor would be needed. Also, if the ear bud I have fails, we would be out of luck.

It seems to take a 2.5 mm male, but when I bought a 2.5mm to 3.5mm adaptor, I only got the faintest bit of sound.

Here is a slightly blurry photo of the male plug that corresponds to the jack I need to plug into:

It appears to have four sections along the pole, including the tip. It only serves a single earbud.

I would like to be able to find an adapter to plug into the jack so that I could get sound in a standard 3.5mm earphone that serves at least one ear. Mono sound in both ears or just in one ear would be fine. My mother is profoundly deaf in one ear and would be using this mostly with her good ear.

Maybe someone can tell me what I have, and what I need.