Recently my son broke my front USB ports on my machine so I simply ran and extension from the back to plug in my USB Wireless adapter.

Windows reports that it found new hardware, but the hardware is not really new since it was already installed. I've noticed this when plugging things into different USB ports, but do not understand whey Windows thinks the hardware is new.

The problem is that the installation was kind of lengthy and required some kind of special sequence that I forgot, and I am not able to locate the installation disk.

What can I do now?

I fear that Windoze in it's infinite wisdom keeps track of what port the device is plugged into. Your one hope is that drivers are installed on the system so if you go through the installation steps it may retrieve them from the hard drive. If not you will have get the installation software from a disk or a download

"but do not understand why Windows thinks the hardware is new"
I think Windows has a different branch of keys and sub-keys in the [Local Machine\Hardware key in the] registry for each particular hardware combination/configuration it has detected [complete with drivers installed].
When you boot the PC, Windows detects which set of hardware is connected, and in which configuration, and uses the appropriate registry hardware branch and set of drivers to suit, and all is well and works as it should.

You could have all sorts of hardware combinations...
If you were to disconnect certain items and reconnect others, then...
Just so long as Windows had met this combination at least once before and had been properly set up to use it, then it would have no problem instantly making the necessary changes/adjustments.
The only possible problem is the 1st time it sees this situation, and making the necessary changes [installation of drivers].

"I am not able to locate the installation disk"
This is why I like to copy the installation file sets [for Windows and other major programs like MS Office] to a partition [other than C:] on the internal HDD, and leave them there permanently.
Then when Windows [or Office, or whatever] needs a file it automatically fetches it from there in an instant [doesn't ask the user to supply the CD in a drive with a letter that's the same as the letter of the optical drive when the original installation was done], and the job's done in the blink of an eye.
That drive letter might no longer be available for use with an optical drive; it might have since been allocated to some other device, and then you have a problem that takes ages to fix [if you EVER succeed in fixing it].

"What can I do now?"
You might try telling the driver installation wizard to search the HDD for a suitable driver [".inf"] file.
Before that you could try searching your C: partition...
a. For all inf files [*.inf] to see if the inf file for the correct driver is there.
b. For all files including key text associated with the driver to see if any likely inf files are found.

If you find the correct inf file you could point the driver wizard at that file.