|What does it show in DIsk Management ?|
That's selectable on the left when you click on Computer Management, or run the command I supplied.
You have not told us where the partition is.
Is it on an internal hard drive,
or on an external hard drive,
or on a memory card,
or on a USB flash drive ??
"If I click new partition, can I just assign the old drive letter and then the system will recognize it?"
That depends on where the partition is, and whether the drive letter you want to use is presently assigned to something else.
If the drive's space is showing up in Computer Management / Disk Management, Windows is already detecting the space on the physical drive, but it does not show up in My Computer or Windows Explorer until Windows recognizes that the partition has been software partitioned using some method Windows recognizes (NTFS or FAT 32 or FAT) and has been formatted, AND a drive letter has been assigned to the partition.
You can't make a new partition on a USB flash drive or on a memory card in Windows by default.
You can make a new partition on an external hard drive or an internal hard drive.
In either case, if you know you had data on the partition previously and you don't want to lose that data, you should NOT make a new partition (if that's possible) until you have tried using a program that can repair the data damage that is causing Windows to detect the drive's space as unallocated.
You assign a drive letter while making a new partition.
In any case, if the drive letter you want to use is already being used, you can change the drive letter assigned to anything to a different available drive letter, C thru Z, in Disk Management, EXCEPT that you can't change the drive letter assigned to the partition Windows itself has loaded from. You can change the drive letter of something that is using the drive letter you want to use to some other available drive letter to free up that drive letter.