I think that you both could work on the trust issue you have which has been created by her behavior, together with the aide of a self help book. If you purchase a self help book(s), it is important to designate a time to do this work on a regular basis when the two of you will not be disturbed along with keeping a journal to write down your thoughts and reactions to what you read and discuss. Both of you should agree on how much time you want to spend together working solely on this and agree to how often you want to work on it, for example: once a week for 30 minutes, twice a week for 30 minute, etc. Be sure to keep the amount of time designated for this at a comfortable level for both of you so that neither of you dread doing this. The above is a healthy process that can strengthen your platonic relationship. A book for you to consider can be found below:
Book 1.It is possible that she is telling you about her lying and apologizing to you for both reasons: to get closure as well as to smooth things over should the two of you become involved in a relationship in the future- but there's no way for me to know for sure. As far as the current situation that you have shared with me, I think it is best for you to keep your relationship with her strictly platonic for your own mental and emotional well being, especially since she is back with her ex and is not 100% available to you. If she should break up with her ex, then you could take it from there, but for now you are better off keeping things strictly platonic with her from this point forward.