Can anyone give me a concise summary of Martin Luther's "Two Kingdoms" theology, and also provide some references back to his original works on the subject?
Without having understood the doctrine of the two kingdoms, it's not possible to understand Luther's theology in its full extent.
It's a misunderstanding to think that by the justification by faith alone all laws have lost their importance. As short as possible: a Christian lives under two completly different kingdoms, where the one is "God's kingdom under Christ"1 – here he lives by faith in the righteousness coram Deo2 – and the other is the world's kingdom – here he is man or woman, young or old, lord, servant, peasant, citizen etc.3 and has to do good works.
Accordingly, the human being is both: Christian person and secular person.
You see: the accusation, that Luther's justification by faith alone shrinks the room for the works that have to be done by us, is a misunderstanding. Especially Luther's theology shows people, how overfull of opportunities for good works the world and the way, which they follow in the world by faith, is.
1: "Gottes Reich unter Christo"
2: before God
3: WA 32, 440, 2
4: "Christperson" and "Weltperson"
I found an article that discusses the subject in a clear and concise manner. The opening paragraph sums up the problem:
The question of the two kingdoms is one of the most pressing and delicate in contemporary religious and theological thought. No other aspect of Luther's theology has been so fiercely attacked as this doctrine. Where Luther drew a clear line between spiritual and temporal authority, and expressly emphasised that under no circumstances should these two realms be confused, this has been interpreted as if he had thereby opened the door to the secularisation of society and given a completely free hand to the State. Some critics have gone so far as to see in this doctrine the ultimate root of the National Socialist ideology, while even a theologian like Karl Barth has sharply criticised Luther from a similar point of view.
Journal of Lutheran Ethics - Luther's Doctrine of the Two Kingdoms: https://www.elca.org/JLE/Articles/931
However, the material is copyright protected and so I can only suggest that the OP read the article.