THIS ANSWER ASSUMES DELETING THE INSTANCE IS OK.
ie: all your data will be gone and that is okay.
I was having the same problem, after upgrading my SSMS.
sqllocaldb i.\MyCustomInstancesqllocaldb dLocalDb instance ".\MyCustomInstance" does not exist!sqllocaldb i .\MyCustomInstanceWindows API call "FileTimeToSystemTime" returned error code: -2147024809.In order to get rid of the offending instance I had to create another MyCustomInstance which I guess will overwrite what's already there, and now you can delete it
sqllocaldb c MyCustomInstanceLocalDB instance "MyCustomInstance" created with version 11.0.sqllocaldb d .\MyCustomInstanceLocalDB instance ".\Octopus" deleted.Then, start the instance and share it.Imperative you start the instance first.
sqllocaldb s MyCustomInstanceLocalDB instance "MyCustomInstance" started.sqllocaldb h MyCustomInstance MyCustomInstancePrivate LocalDB instance "MyCustomInstance" shared with the shared name: "MyCustomInstance".Now, when you need to connect, you connect with (localdb)\.\MyCustomInstance