The Luftwaffe didn't have the capacity to mount anything resembling the Blitz in 1944. They simply lacked the bombers, especially without stripping the Eastern Front.
Perhaps with Jabos, similar to Bodenplatte, but it would pretty much be suicide on part of the Luftwaffe. If I recall correctly, the Allies sortied around 7000 planes on D-Day, while the Germans would struggle to muster 1500 of all combat types.
Assuming they were willing to destroy themselves as a combat force (when in reality they held out until Bodenplatte as an organized and effective (relatively speaking) fighting force), and expend all of their fuel stockpiles, they could likely sustain perhaps two weeks of operations. The question would be whether their squadrons became combat ineffective due to losses and damage (and bear in mind that some units could have a ready rate as low as around 65% due to lack of spares anyway), or they ran out of fuel for sustained high-tempo operations first.
In the east they held out a bit longer due to superior aircraft and force concentration, but by spring of 1944, the Luftwaffe was on its last leg, and it was in the east where that leg was standing.
Realistically, the Luftwaffe needs at least a 1941 POD to maintain combat effectiveness into 1944. But basically they need to abandon bomber development save for a few select models. Only the Do 217 and Ju 188 should be developed, everything else needs to be scrapped.
Second, the Ju 87 needs to be scrapped in favor of something more survivable, be it an attack aircraft or just retooling the Junkers lines to 190 production and using Jabos. But those death traps are limited in effectiveness, and inefficient use of resources.
Third, except in the east, the Germans need to adopt a strategy of air denial. Unless a major operation is underway, the Luftwaffe needs to carefully preserve and husband its strength, so that it has that strength to use in moments of need.