The only function of an extended partition...is to contain any number of smaller partitions, which are called/designated logical partitions. Or, to put it another way, the only way to have any logical partitions...is to create an extended partition first.
An external USB hard drive is normally not formatted as an extended partition. There is no need for it, nothing to be gained, since an external hard drive is used as one big storage cabinet.
External hard drives may be formatted as either FAT32 or NTFS file system, with the partition created being a primary partition.
If you have an external drive that does not reflect a drive letter in Disk Management, please post all the info reflected (in Disk Management) for that drive.
If the drive had previously been properly recognized and it now reflects nothing but unformatted space...either your enclosure or the drive inside the enclosure...is probably going bad.
The actual hard drive can be removed from the enclosure.
I...not necessarily you...I would remove the drive from the enclosure and attach it directly to the motherboard temporarily...to see if the motherboard/BIOS/Windows can properly recognize it if directly attached.
In the event that the drive itself is failing, your logical recourse is to attempt to use data recovery software of some type to reclaim/recover data files on the drive.