For a girl who has abandonment issues due to neither of her parents showing her the affection she needed growing up, there are worse things that could happen than being tied up like that so she won't leave her lover.
The codependent desperation is palpable but keeping someone tied up at your place when you've been accused of forcing them into a relationship is not going to look good if Mom presses further. Io is basically in position where she either publicly comes out on her own or watches on as Misa potentially faces prison time.
Ah yes the solution to all problems, good ol' BDSM. But in all seriousness, Misa doesn't even have to tie Io to get her to stay, Io is already extremely attached to her.
If she's an idol that wants to defend herself, couldn't she go live on her social media and just show the conversation to take down her mother and the rapist she calls a manager?