I see your point – I would have expected to see knew written there, to keep the past-tense phrasing consistent.
If you are sure your original quote definitely said know and not knew, then I would say it is either a mistake by the writer, or a deliberate attempt by the writer to describe a quirk in the old lady's accent/dialect. Personally, I think the mistake is more likely.
Went is just the normal, past-tense form of go. Go is one of English's many irregular verbs, in that it does not follow the regular rule of adding -ed to the end to make it past-tense. For example, you don't say "He goed to the shop," you say "He went to the shop."
Would go is a special past-tense form that is used to describe a habit, routine or other regular occurrences that someone used to do. In this circumstance, you can say "I would go...", "I'd go..." (the I'd is a contraction of I would) or "I used to go...".
You mostly use went to describe actions that you did/places that you travelled to once or infrequently, rather than things you used to do routinely:
- "I went to the shop today. I bought bread and eggs."
- "I used to go to the shop for my mother. I would buy bread and eggs."
- "I used to go to the shop for my mother. She would ask for bread and eggs."
Note that there is another use for "I would go..." (and, by extension, "I'd go..."), but that is for describing what you would do in a particular situation, e.g. if you were giving advice. This is not the form that your quote is using, but you will see that I use it in my advisory statements to you.
Given the irregularity of the go verb, I probably wouldn't use the "I would go..." form that is used in your quote, unless it was already contextually established that I was talking about a past habit. In your quote above, this context is given by the sentence starting "I used to..."
However, I would probably always choose to say "I would not go..." (or the contraction "I wouldn't go...") if I were talking about an action that I deliberately did not do (or a place I deliberately avoided). I would not usually say "I used to not go...", although you may hear this in casual speech. You definitely wouldn't say "I did not went..."—it is always "I did not go..." (or "I didn't go...").
Also, in your quote, would go has been used in two places – "when you would go out" and "I wouldn't go to sleep". The first usage is talking about you leaving the house to go to places (socially, e.g. to a bar). The second is just another irregularity – you do not "do sleep", you "go to sleep".
Finally, I am comfortable replacing would go with went in the "when you would go out" part of the quote, because the old lady is really describing her routine (as a reaction to your actions), not your routine of going out. You can replace it in the other parts of the quote, but you may need to restructure the sentences a bit to make the context obvious:
- "I used to stay up every night when you would go out with your friends"
- "I used to stay up every night when you went out with your friends"
- "I would stay up every night when you used to go out with your friends"
- "I wouldn’t go to sleep until I heard your door close"
- "I would stay awake until I heard your door close"
- "I only went to sleep when I heard your door close"