What is the black slag formed when sterling silver is melted with borax?
"Indestructible" by what means exactly? How did you try to "destroy" it? Provided that sterling alloy is an alloy of silver and copper, the first guess is that your slag is primarily a copper (II) oxide. Depending on the actual composition of the alloy, it may contain other 3d-metal oxides. The formation of borates (not of silver, but of less noble metals) cannot be excluded, for borax is frequently used as a constituent of the high-temperature fluxes for spectroscopic and calorimetric applications. To what extent the borates are formed depends on the temperature and duration of your melting experiment.

You should be able to dissolve your slug in a boiling mineral acid of your choice. Personally, I'd try the $\ce{HCl}$ solution, or $\ce{HNO3}$ if you do not care that silver dissolves as well.

The following may be irrelevant to the question asked, but, based on my experience and some textbooks, people first try to purify silver and then melt the pure metal, not the other way around. The methods of purification are aplenty, but most of them are based on dissolving the silver-containing alloy in $\ce{HNO3}$. Then $\ce{AgNO3}$ can be transformed into $\ce{AgCl}$, which, in turn, can be reduced by metallic zinc in acidic environment or formaldehyde - in basic.