since she is remorseful bout betraying you when she's
lonely.
i suppose she is an emotion gal, and needs lots of company.
she really needs to lean onto you for support. just give her all
the emotional support and she needs from you.
if you want to speak to her on clarifying things which your not
really sure with attempting to make the relationship work again or
not, speak to her only when she is there leaning on you, when she
is comfortable, no eye to eye contact like questioning, body
contact, perhaps a slight hug then ask her, if she gives you a
reasonable ans, then hug her warmly or just kiss her on her
forehead to give her the reassurance.