[Dachshund Tim]

Here's Buhler's website list of small Motors or contact them to see if they could help.
You could also try this guy TrainCraft by Klaus - LGB Locos, LGB Parts

 

[Dachshund Tim]

Did you try contacting Buhler directly here in the U.S.
Buehler Motor, Inc.
1100 Perimeter Park Dr., Suite 118
Morrisville, NC 27560
Tel.: +1 919 38 0-3333
Fax: +1 919 38 0-3256
e-mail: sales@buehlermotor.com
Tech/Sales Office Detroit:
27220 Haggerty Road, Suite D7
Farmington Hills, MI 48331
Tel.: +1 919 38 0-3333
Fax: +1 919 38 0-3256
e-mail: sales@buehlermotor.com

More looking on Buhler's website came up with more contacts, here's a link for the US.
Distribution partners

Here's a few send inquiry.
Brushed motor - 1.13.021.602 - Bühler Motor GmbH - DC / 24 V / permanent magnet
1.13.021.602 - Onrion LLC
Buehler Motor 1.13.021.602 - buy on famaga.com

If you find a motor for your Mikado here's a link from Raymond about the motor overheating.
LGB - Mikado - Repair & Modification Page

 

[Dachshund Tim]

Here's the part number from an old Only Trains listing LGB Model Trains, LGB Locomotives, Garden Train Sets, G Scale Model Trains,: E161892 Buhler Motor for LGB Mikado Steam Locomotives

You might be able to pull the worm gear off the old motor and reuse it on the new motor if you're careful. I checked my Mogul and Stainz spare motors and it looks like they are pressed on, no signs of solder.

I've sent inquiries to websites like I listed looking for spare motors and received responses on whether they have it or don't. The distribution partners may steer you to someone who carries it so try calling them.