As far as I am aware, a Dadant deep is 2" deeper than a Langstroth (or the same as a Langstroth Jumbo, although with slightly deeper frame spacing). Moving to a Dadant will, therefore, give you the equivalent of an extra 2 Langstroth frames (approximately). This is far less than the 10 extra frames you would have in a double Langstroth so it wouldn't give you the extra space you are looking for. Am I missing something?
Yes, you are missing two critical pieces of information. I am setting up Square Dadant depth boxes which are 18 5/16 interior measure both directions and 11 5/8 inches deep. I also use 31mm frames. I can get 14 frames in one box. A Langstroth 9 1/8 frame has 7100 cells of capacity and in a double box gives 142,000 total cells. A Dadant 11 1/4 frame has 9000 cells of capacity and with 14 frames gives 126,000 total cells. This is not the end of the story though, have you noticed that queens rarely lay in the corners of frames? Fourteen of the Dadant depth frames have 24 less corners than 20 Langstroth frames. This should translate into about 5000 more usable cells in 14 Dadant frames.
This begs the question, why do I use 1.25 inch frames? I've used them since 1977 and found some significant benefits. The first is that bees build up measurably faster in spring on 1.25 inch frames.
Another item you might ask is how much does it cost to set up a Square Dadant depth hive compared to two Langstroths. I can fully equip a Dadant depth square hive for $51 vs roughly $52 to equip two Langstroths. These numbers will vary quite a bit depending on source of materials. I am factoring in $1.75 X 14 for foundation, $1.00 X 14 for my time and material to build the frames, and $12.50 to purchase the 11 5/8 deep box. For the Langstroths, I am factoring in $1.25 X 20 for foundation, $.80 X 20 for frames commercially produced, and $11.00 for the 9 5/8 deep box.
Here are comparison numbers for several sizes of frames including Dadant deep frames.
Here is 4.9 foundation cell counts
11 1/4: 10513
9 1/8: 8307
7 1/4: 6360
6 1/4: 5322
5 3/8: 4413
Here is 5.1 foundation cell counts
11 1/4: 9705
9 1/8: 7668
7 1/4: 5871
6 1/4: 4912
5 3/8: 4074
Here is 5.3 foundation cell counts
11 1/4: 8986
9 1/8: 7100 (* this is standard Langstroth deep foundation cells per frame)
7 1/4: 5436
6 1/4: 4549
5 3/8: 3772
Here is 7.1 drone foundation cell counts (* The hive and the honeybee gives 6.67 for drone combs!)
11 1/4: 5007
9 1/8: 3956
7 1/4: 3029
6 1/4: 2535
5 3/8: 2102