Letting her come to you at her own pace could mean an indefinite amount of time waiting.
If you want to speed things up with her, it may be best to cut contact.
If she cares enough about the interaction, this could force her to get out of her comfort zone and try seeing you on a one on one basis.
She may be cold when you reach out to her because she is being guarded.
She may be partly worried that you are only reaching out so as to ask her out and she may not welcome this given the protectiveness of her household.
She may not be cold when she reaches out to you because the side of her that just wants your attention may be the most prevalent emotion she is experiencing at that moment.
That emotion may trump her anxiety that you may use this opportunity to try asking her out or asking her to hang out.