On the website you are linking a specific approach is explained on how to manage a survey to be somehow anonymous without using the anonymous mode of Limesurvey. So if you followed the instructions there, you might be fine.
If you have set the survey to anonymous on Limesurvey, then you are screwed...
In the tutorial it specifically mentions that you do not set the survey to anonymous on Limesurvey. The anonymity is achieved by the use of the an anonymous email service:
Participants must not sign up for the survey using their actual email address. Instead, they should go to a third-party email forwarding site like
notsharingmy.info
. For the participant:
If you explain a little bit what you did and how your set up and invitation process was, maybe we can see if there is something that can be done. But I have very little hope.