The broken wire is a nicrome resistor. I have heard that these can be upgraded to modern resistors. This seller (no affiliation or recommendation) has these available:
https://www.ebay.com/itm/Lione
You should also note that the 1044 is a relatively small transformer only able to deliver 60 watts or so and comes up short of delivering the traditional 18VAC used in today's command systems and so it may be that there is nothing really wrong principally with the transformer other than being overloaded.