Making toad in the hole with duck eggs
Toad in the hole batter, which is the same as yorkshire pudding batter, uses eggs for lift and partly as the binding agent. Lift comes from expansion of gases, including water vapor. Egg white is 90% water, yolk is somewhere around 40% water. 

A large egg is usually somewhere around 60g out of the shell. With a hen's egg 2/3rd of that is white, so call it 44g of water in total (36 from the white, 8 from the yolk). A duck egg is usually about 1/3 white, so about 34g water in total (18 from the white and 16 from the yolk). So using duck's eggs you get 20g less water overall in a doubled delia recipe. Given that there's 260ml of water and milk already this isn't much of a difference, maybe about 6-7% total water content. This doesn't sound like much but it can make a big difference. 

Fat-wise there's about 10g more fat in 2 duck eggs than hen's eggs. A hen egg yolk has roughly 5g of fat in it, and a duck egg double that. 

Protein-wise white has a higher percentage of protein (7%) than yolk (5%), but that's not much so not likely to be causing issues. 

It's probably the combination of less water and more fat which is causing your problem. Less water and more fat means less expansion, less expansion will mean it takes longer to cook and has a flat texture as you describe. 

As for the remedy I think you should definitely try 3 whites and one yolk, that should bring the balance back just right. 

One trick recommended by a few chefs is to wait until your toad in the hold or yorkshire batter is fully expanded, then open the oven for a few seconds to let the water-saturated air out. This will help the batter to crisp up.