I've encountered a bug in which the quest marker shows up properly for Danica, and is indicated on my screen and everything, but when I talk to her there is no dialog option that would allow me to complete the quest.
I have encountered a bug that prevents me from opening a Dialog with Igrod the Younger and it is preventing me from recieving the quest to deliver her letter to Danica.
(126.96.36.199 06:05, February 24, 2012 (UTC))
Can one marry Idgrod the Younger without any console edits or mods ?
No, she's unmarriable. There is, however, a nonessential save for the 360 out there by a guy named Joe Cobbs or something like that, just Google 'nonessential skyrim save xbox' and it should be the first link. He's removed the nonessential tag from pretty much all vanilla NPCs (not including DLCs) and I think she might be one of the marriable options.