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I got stuck on this for years until I noticed something simple: "quelque" on its own generally comes before a noun or adjective ("quelque temps", "quelque peu"), whereas "quel que" is always followed by a verb in the subjunctive and you can actually split it into two words without it sounding wrong ("quel que soit", "quelle que soit ta réponse"). Now when I'm unsure, I mentally test whether I can say "n'importe quel" instead - if yes, it's two words, if not, it's one.
"Quelque" is a determiner or an adverb that means a certain quantity or a certain number - like "quelque chose," "quelques jours." "Quel que" (two words) is a conjunctive phrase you use with the subjunctive to say "no matter which" - like "quel que soit ton avis, je fais ce que je veux." The big difference is that with "quelque" you're talking about an imprecise quantity, whereas with "quel que" you're saying "it doesn't matter which one."
The thing that changes everything is checking whether you can replace it with "no matter what" - if that works, you're in "quel que" territory (two words). For example, "quel que soit ton avis" → "no matter what your opinion," that works. Whereas with "quelques amis" you can't say "no matter what friends," so that's "quelque" as one word. Basically, "quel que" expresses indifference (it's a conditional or concessive thing), while "quelque" is just a vague quantity.
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