using iMac 27 -broke screen and using external monitor

the system is now running so slow like it's a pentium one or 486 system-- mine is a i7 Quad 4.0 with 32 gig ram . I tried reloading the drive with basic catalina and no other S.W. and just opening the OS preferences is spins and Wait - wait ..

What kind of internal storage does the 2015 iMac have? Mechanical drive? Apple's hybrid Fusion drive? Apple's SSD? That is important when the old "hold it or heave it" conundrum arises.

I'm all for keeping old Macs working at age-appropriate tasks, but what you describe sounds like that iMac has more issues than just a broken screen. That kind of "slow" in iMacs generally indicates a failing mech or Fusion drive. As 2012 and later iMac are not designed to be opened, servicing or replacing the internal storage is a complex and delicate matter best left to pros. And I've talked to some pros who will not work on sealed-case iMacs, largely because they have to charge twice as many labor hours at, in my region, US$65-80/hour.

It's time to cut bait and buy a replacement Mac. You are trying to breathe life into an obsolete computer that has zero value and is running slow which likely means a failing hard drive.

You have a LOT of options open to you, trying to save your existing machine is the poorest one. You can buy a new Mac that will far outperform your current machine, be able to run current versions of Mac OS and get a warranty with it. If you feel you cannot afford it, see if you qualify for an Apple Card and get up to 24 months of free (no interest) financing. You can select from another iMac, a Mac mini, a MacBook Air, MacBook Pro, or a Mac Studio. In short you have a LOT of options open to you. You can even get an Apple refurbished Mac that can suit your needs, these have absolutely NO downside over buying a brand new machine. Refurbs are available in many countries, just go to Apple's home page in your country (I'm in the US so that would be apple.com) scroll to the bottom of the page and click the Refurbished Store. If you don't see exactly what you want check often as inventory changes very quickly.

Not sure about the Late 2015 iMac's, but do I know that disconnecting the cables from the display panel and then plugging in an external display works with some iMac models. If fact, the external display will become the main display, instead of being Mirrored or Extended to a broken internal display.

The problem that you might have with some models, is controlling the Fan speed if there is a temp. sensor integrated into the display cabling.