Will the diskette work in another computer? If so, then there is an issue with the FDD.
Verify that the FDD cables (I assume this is an internal FDD connected to an FDD header on the motherboard) are oriented properly and firmly plugged in. The side of the FDD cable with the white (or grey) wire represents Pin 1. If your FDD cable is keyed, this won't be an issue as it will only connect one way.
If connected properly (you have to check both ends of the cable (on the FDD and motherboard) and it still won't read a diskette that you have verified as good, then the FDD is either dirty or defective. You can use a can of compressed air to remove any dust/crud build-up.
If defective, you can replace. If you replace, you might want to consider a USB FDD that you can move from system to system.