If Instagram asks you for a code you can't type into Storrito's connect dialog, it's worth checking which kind of code you're looking at. Instagram sends two different things that look similar but aren't interchangeable: a login code (a normal 2FA code, generated for a routine login) and a recovery code (issued when Instagram thinks the account needs to be recovered, not just logged into).
Storrito's connect dialog only handles the login flow. It has no field for account recovery, because recovery is something only you can complete directly with Instagram.
This usually happens to accounts that don't have two-factor authentication (2FA) set up yet. Without 2FA, a login attempt from a new device or app can look suspicious to Instagram, and instead of a normal login code, Instagram drops the account into its account-recovery flow.
Trying again and again, especially with different passwords, makes this worse. Repeated failed attempts read as suspicious activity to Instagram and can push the account further into a recovery-only state, where it keeps asking for recovery steps instead of offering a normal login.
If the account is shared by more than one person, an authenticator app is the more reliable option, since anyone with the app can generate the current code without needing access to a single phone number.
A login code is short (usually 6 digits) and is offered as part of a normal sign-in. A recovery flow instead asks you to confirm identity details, prove you own the account, or wait out a review period — if that's what you're seeing, you're in account recovery, not a login, and Storrito's dialog can't take you through it. Recovery has to be completed with Instagram directly, after which you can return to Storrito and connect normally.
See also Connection Methods for the general connect flow.