Dear @Sam Sam,
The employment agencies (Agentur für Arbeit and Jobcenter) can finance courses for people who are unemployed, who are in danger of becoming unemployed or who wish to retrain. These courses are commonly called Umschulungen or Weiterbildung. They have to be certified in order for the authorities to pay for them. There is also not an automatic entitlement to a particular course. Caseworkers assess each application individually to see whether funding a particular person to do a particular person is worthwhile. The fact that you also have a qualification in this field would help your case.
I looked through the site of the German Gemmological Association (I did not find one called the German Gemstone Institute). I did not see anything there about certification for financing via the Jobcenter or Agentur für Arbeit. I think you should call and ask them about this, they seem to have extensive information in English.
Normally for a person who is working the responsibility for training would be with the Agentur für Arbeit. As you are on sick leave, I presume you are in receipt of sick pay (Krankengeld). I’m not sure to what extent the Agentur für Arbeit will assist you while you are unable to work.
What you can try, however, is to speak with the German pension authority (Rentenversicherung) about professional rehabilitation. The Deutsche Rentenversicherung is responsible for paying those who are unable to work, for example due to advanced age or illness. They also, however, help people to re-enter the workforce via professional rehabilitation:
https://***.deutsche-rentenversicherung.de/DRV/EN/Leistungen/leistungen.html
There are special advisory services called WUTB to assist individuals with health challenges and disabilities with such questions as yours, I think it would be a good idea to contact one in your area:
https://***-teilhabeberatung-de.translate.goog/artikel/was-sind-die-eutb-angebote-und-was-machen-sie?_x_tr_sl=de&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp
Best,
Éanna