For extraction, you'd want to use different solvents that have different dissolving properties - usually something like water, ethanol, DCM, acetone, MEK, methanol, toluene or whatever.

For strontium, it looks like it's relatively soluble in short chain alcohols (methanol/ethanol) compared to the other two, so you'd crash out the potassium perchlorate by dissolving the mixture in water, then reducing the temperature to cause perchlorate to drop out of solution, then mix in a moderate amount of methanol to crash the potassium nitrate out, being left with a reasonably pure strontium nitrate, that you could then hot filter and recrystallize in anhydrous methanol if you wanted >90% purity. One or two rounds of recrystallization will leave you in the high nineties, probably above 97%.

This is a classic chemistry workup kind of problem and there are interesting engineering challenges embedded in it.

Of course... practical people just buy technical grade strontium nitrate and make fireworks out of it directly, as the article says.