"The television that stands next to the new table is expensive."
Translation:Az a televízió drága, amelyik az új asztal mellett áll.
I am a native speaker of Hungarian. If you used 'ami', I would hardly notice, so don't worry. I myself probably mix these two sometimes. You'll find a very detailed explanation here (in Hungarian): https://www.nyest.hu/hirek/ami-es-amely The gist of it has something to do with this: you need a non-animate object/thing in the first clause (as opposed to a person, in which case you'll use 'aki', 'akit' etc.) to which 'amelyik' refers to, plus the same non-animate thing should be a nominal (basically a noun) as opposed to a pronoun. If it is the whole clause to which you want to refer back, it is always 'ami'. If you have a (demonstrative) pronoun in the first clause, again use 'ami'. There are some borderline cases (e.g. 'Az jó, ami van' vs. 'Az a jó, amelyik itt van' [standing for: 'Az a [dolog] jó, ami itt van.']), but even a native speaker can get confused about them, so again: don't worry.
Amely/ami and amelyik are singular, amelyek/amik is plural.
From my understanding, amelyik is used when you are talking about something specific, while amely/ami is used when you are talking about something vague.
Amely and ami mean the same thing, the second form is just shorter. Same with amelyek and amik.
I think you're right here. The way i think about it that helps me sometimes is to think about what the question words mean if you were to remove the "a". "Melyik" usually means "which one" and is used in relation to items that can be counted or specifically identified. If you're talking about something, where you could substitute "the ___ one" for the noun, then "amelyik" would be appropriate.
In this case think about starting with "the tv is expensive". Then imagine there are several tvs sitting nearby that might have been referred to, so you ask "which tv?" ("Melyik tv?") In english we specify which one by saying "that one" or "the one that ...". In Hungarian we respond by mirroring the same question word that was used or implied: "az a tv amelyik ..."