Learn Russian: Мой брат проспал в кресле всю задержку. - My brother slept in an armchair through the whole delay.

It often does! However, when you pair a verb that has the про- prefix with an accusative of time (like всю задержку here), the meaning shifts to "spent [duration] doing [action]". So here, it means he spent the whole delay sleeping. If the sentence was just Мой брат проспал, it would mean he overslept.

Why is the duration всю задержку in the accusative case here?

This is the "accusative of time" working together with the про- prefix. To show that an action spanned an exact, continuous block of time, Russian uses the bare accusative case without any prepositions (no need for words like "during" or "for").

Why do we say в кресле (in) rather than на кресле (on)?

In Russian, an armchair (кресло) is treated as an enclosed, plush space that you sink into, so it requires the preposition в. This contrasts with a regular, hard chair (стул), which is treated as a flat surface and takes на (на стуле).

Is the word задержку related to the verb держать (to hold)?

Exactly. It comes from the verb задержать (to delay / to detain), which is built from the root -держ- (hold). So задержка is literally a "hold-up" or delay. Here it is in the accusative case (задержку) because it acts as the time duration.