The structure you learned (no sólo A sino también B) is fine, but in this case, when B is not a noun phrase but a complete proposition (a subsentence with its own verb), then it requires que before it (DPD, sino, 2.1). If B were a noun, you wouldn't need que.
This is what you learned:
no sólo [(pasa) algo]A sino también [otra cosa]B
Note that A can be a noun phrase alone (algo = "something") or it can have a verb (pasa algo = "something happens"). It doesn't matter; the structure works the same for the first part.
But when the second part B is a full proposition (a subordinate sentence with a verb), you have to add que:
no sólo [(pasa) algo]A sino que también [pasa otra cosa]B
Also, see that también no is wrong; in Spanish también + a negative is rendered as tampoco:
No sólo no puedo ver esto sino que tampoco puedo escuchar esto.
The natural thing to do in a sentence like this, where the object of the two verbs in the two parts of the sentence is the same, is to replace the latter instance with a shorter clitic pronoun:
No sólo no puedo ver esto sino que tampoco puedo escucharlo.
Finally, in this case you could shorten the sentence even more and get rid of the que by making both parts of the sentence subordinate to the verb puedo:
No sólo no puedo ver esto sino tampoco escucharlo.
This works, I think, because A = ver esto and B = escucharlo; they both function as noun phrases and the main verb stays outside. (If this is too complex for you, forget it for now; just remember the general rules stated above.)
