Can you try the following:
- Install SQL Server Express.
- Create a new database with the same name, so a new LDF file with the same name will be created.
- Shutdown SQL Server Express*.
- Replace the new LDF file with the existing one.
- Restart SQL Server Express.
This may work, although it wouldn't surprise me if it didn't because the new database doesn't have the same structure.
* It's also possible to detach the files from the database and reattach them but that requires to call a duo of stored procedures. This way is easier.