So, from MH:
01.09.1939 - 31.12.1944
1,965,324 (KIA) + 1,858,404 (MIA) + 5,240,000 (wounded) = 9,063,728
+ unknoun figure for 1945 (most disastrous year for Gernans like 1941 for Soviets)
and
+ German's satelites
1,965,324 is the number of dead for all causes (including died of wounds, diseases, accidents etc.), not just killed in action (which is certainly a large subset). It is taken from the Wehrmachtverlustwesen records (OKW/AWA/WVW(V)).
I have a more detailed breakdown of the numbers from the Wehrmachtverlustwesen not for 31.12.1944, but for 30.11.1944, when the number of dead was 1,911,300 (this is appended to Hermann Jung's dissertation on the Ardennes offensive). This included 1,725,616 killed or died from wounds received by enemy action, 175,901 killed through accident, disease and suicide and 9513 executed by military justice for crimes committed. Breakdown to branches was as follows: Army+W-SS: 1,709,739, Navy: 51,774, Air force: 149,787.
Additionally the figures include all fronts from 1939 to 31.12.1944, e.g. 66.266 dead and 3.218 missing in the West prior to 30. May 1944 and 54.754 dead and 338,933 missing in the West between 1 June and 30 November 1944, as is listed in M.-H. as well (3 pages later).
Finally, Zetterling and Frankson in an article in the Journal of Slavic Military Studies look at a similar set of figures from Wehrmachtverlustwesen, which however does not include the period 1.9.1939-21.06.1941, so the figures are slightly lower. The figures presented there show 1,239,425 army and W-SS soldiers killed in action, including 1,091,962 on Eastern front, the rest in other theaters, 219,087 air force soldiers killed in action (including Eastern front: 53,469) and 34,998 navy soldiers (including Eastern front: 8,039). Additionally 278,480 army and W-SS soldiers died from wounds received by the enemy and 134,536 died for other reasons such as accidents and diseases. All these figures covering the period 22.06.1941 - 31.12.1944.