How does he hold any of the cards? The military is fresh off of it's victory in France and the SS has a large militant arm at it's disposal, along with the Gestapo. The Luftwaffe doesn't have any powerful ground forces in 1940 capable to carrying out serious power play. Any attempt to seize more power than he's able to negotiate will result in his rapid removal from power.
The SS isn't that big at this time. Besides, Himmler was not well liked, certainly not by
Wehrmacht leaders. I imagine the army would side with Goering and push for a conservative-nationalist type of regime with Goering as the new
Ersatzkaiser. I imagine the NSDAP would become a bit of a dead letter from then on.
As for the war:
- Goering was initially opposed to Barbarossa and probably won't invade the USSR ITTL, leaving Stalin to happily ride out the war as a neutral.
- BoB still goes awry and Goering doesn't go for Sealion.
- Mediterranean strategy to support Germany's Italian allies, leading to bigger success there.
- From there: keep the US out at all costs (possibly through renouncing Japan and the Anti-Comintern Pact) and hold on until the British public gets tired of the war and demands peace.
- TTL's Germany wants 1914 borders + Sudetenland and Austria. Goering can hereby easily offer to establish an independent Polish rump state, and Polish independence is what the war was fought over to begin with (also, the Czechs could be regaining their independence, although as puppets to Germany). The British government might like pre-war Polish borders but that won't happen (especially since Stalin will side with Goering on the matter), and I imagine the British public will have its concerns about fighting for what seems a lost cause.