The Migration Assistant, hereinafter referred to as the MA, is programmed to extract data from a number of different places but the data has to be in specific places for it to be retrieved.

If you choose "From another Mac", MA knows exactly how the data is laid out on that other Mac's hard drive.
Applications in the Application Folder, Documents in the Documents folder, Pictures in the Pictures folder, etc.
You get the idea.

In your case, the data that was "recovered" from the dead Mac is not in any sort of order that MA is familiar with. Doubtless the data is kinda logically arranged, Pictures in the Pictures folder, etc, but is almost certainly not exactly as it was on the original drive.

This is not a problem as you would not have been able to transfer quite a few of your applications from the old to the new so all you have to worry about is the data that you created yourself, Pictures, Documents, Movies, Music and the like.

Ignore the MA and move on from there. Connect the external drive to the iMac and it should mount on the desktop.
Open it up and drag the CONTENTS of the folders, NOT the folders themselves, into the corresponding folders on your new iMac.
Do not bother with any of the applications that may be on that external. You can reinstall those as necessary.

Do you still have the old internal drive? Even though you took it somewhere to have the data transferred, you should still keep the old drive. It's yours and it contains ALL your confidential information. Names, addresses, passwords, Banking information (including usernames and passwords) credit card information. Get it back.

Did the old iMac die from old age or did the drive fail? If just from old age, the drive would still be intact. An external enclosure for it would have solved the MA problem and had you on the way in no time.

Let us know how you get on with the transfer