Take a back-up of the register first. Google this if you don't know how to. Esnure you back-up the whole registry, not just a branch.
Keep the back-up for at least a few weeks until you know this issue is resolved.
I went into the registry, searched for "logmein" and deleted any folders that appeared to be exclusively for logmein.
If the folder did not appear to be exclsive, I delete the keys instead.
There were quite a few entries to delete - probably 20 at a guess. I only felt confident doing this because I knew I had a back-up so that is very, very important to do.
Get help if you need to.
This enabled me to install the latest version, from an old version which did not uninstall correctly.