No, you couldn't.
In questions, the "ikke" will (almost) always be directly after the subject of the sentence. And the subject of this sentence is "din idé" and therefore, the "ikke" is put after that.
Another example could be: "Hvorfor kan han ikke se det?" ("Why can't he see that?").
The subject is "han" so the "ikke" is placed after that :)
Yeah, your example is exactly why I said "almost always" - because (I was not sure, and I actually seem to be mistaken, hang on ... and because) "hvorfor ser han det ikke?" is correct. But "Hvorfor ser han ikke det?" is also correct, although it has a slightly different meaning (the "det" will be more stressed, and it will mean something more like "Why didn't he see THAT?" instead of "Why didn't he see it?").
I am a native Danish speaker, so I can only speak descriptively (I don't know every rule of Danish grammar, and I am guessing most of what I'm saying now), but it seems like this is the case:
O̶k̶a̶y̶,̶ ̶s̶o̶ ̶m̶y̶ ̶"̶i̶k̶k̶e̶ ̶a̶f̶t̶e̶r̶ ̶t̶h̶e̶ ̶s̶u̶b̶j̶e̶c̶t̶"̶-̶r̶u̶l̶e̶ ̶s̶e̶e̶m̶s̶ ̶(̶o̶n̶l̶y̶)̶ ̶t̶o̶ ̶a̶p̶p̶l̶y̶ ̶w̶h̶e̶n̶ ̶t̶h̶e̶r̶e̶ ̶i̶s̶ ̶a̶ ̶m̶o̶d̶a̶l̶ ̶v̶e̶r̶b̶ ̶o̶r̶ ̶a̶u̶x̶i̶l̶i̶a̶r̶y̶ ̶v̶e̶r̶b̶ ̶i̶n̶v̶o̶l̶v̶e̶d̶ ̶a̶n̶d̶/̶o̶r̶ ̶i̶s̶ ̶t̶h̶e̶r̶e̶ ̶i̶s̶ ̶n̶o̶ ̶o̶b̶j̶e̶c̶t̶:̶ ̶"̶H̶v̶o̶r̶f̶o̶r̶ ̶k̶a̶n̶ ̶(̶m̶o̶d̶a̶l̶ ̶v̶e̶r̶b̶)̶ ̶h̶a̶n̶ ̶i̶k̶k̶e̶ ̶s̶e̶?̶"̶ ̶o̶r̶ ̶"̶H̶v̶o̶r̶f̶o̶r̶ ̶h̶a̶r̶ ̶(̶a̶u̶x̶i̶l̶i̶a̶r̶y̶ ̶v̶e̶r̶b̶)̶ ̶h̶a̶n̶ ̶i̶k̶k̶e̶ ̶s̶e̶t̶ ̶d̶e̶t̶?̶"̶ ̶=̶ ̶t̶h̶e̶ ̶i̶k̶k̶e̶ ̶i̶s̶ ̶a̶f̶t̶e̶r̶ ̶t̶h̶e̶ ̶s̶u̶b̶j̶e̶c̶t̶
O̶t̶h̶e̶r̶w̶i̶s̶e̶,̶ ̶i̶f̶ ̶t̶h̶e̶r̶e̶ ̶i̶s̶ ̶n̶o̶ ̶m̶o̶d̶a̶l̶ ̶o̶r̶ ̶a̶u̶x̶i̶l̶i̶a̶r̶y̶ ̶v̶e̶r̶b̶s̶ ̶i̶n̶ ̶t̶h̶e̶ ̶q̶u̶e̶s̶t̶i̶o̶n̶,̶ ̶t̶h̶e̶ ̶i̶k̶k̶e̶ ̶i̶s̶ ̶a̶f̶t̶e̶r̶ ̶t̶h̶e̶ ̶o̶b̶j̶e̶c̶t̶:̶ ̶"̶H̶v̶o̶r̶f̶o̶r̶ ̶s̶e̶r̶ ̶h̶a̶n̶ ̶d̶e̶t̶ ̶(̶o̶b̶j̶e̶c̶t̶)̶ ̶i̶k̶k̶e̶?̶"̶ ̶o̶r̶ ̶
Wait, no, this is also not consistent ... I just realized I have (apparently) no idea how to explain my own mother tongue.
Okay, so my conclusion is: Everything I have tried to explain is true - in some cases, but you will need someone who's better at explaining than I am to explain why the Danish word order is how it is.
I hope I was of help to some extent, and that I wasn't just confusing to listen to :)