The problem with the real thing is there no real way to speed up the deposition of CaCO3 very much - though I seem to get quite a bit blocking the taps at home within a matter of years. I've never heard of making stalactites or stalagmites with sodium silicate and Epsom salts (MgSO4.7H2O) but I can see it might work.
The picture actually shows stalactites rather than stalagmites. If you are really after stalagmites, then making a chemical garden might suit your needs. This too involves sodium silicate. Instructions here from the Royal Society of Chemistry: https://edu.rsc.org/experiments/making-a-crystal-garden/416.article . This, being designed for chemistry teaching, proposes various chemicals that you would need to order specially. But you could also try iron (II) sulphate, greenish but may go a bit rusty-brown, which is sold in garden centres for calcifuge plants, and copper sulphate, blue, which is or was sold, mixed with calcium hydroxide (I think), as something called Bordeaux mixture to control disease on plants, as well as a Epsom salts.