No, I cannot do this.
A card discarded with madness is discarded, but is sent to exile instead of the graveyard. Rule 702.35a.
The complication is the act of casting it from exile for its madness cost. On moving it from the exile zone to the stack, the spell becomes a new object. Rule 400.7
When it resolves, and is put in the graveyard, the triggered ability ...
No, you cannot do that.
Once you cast a spell using the Madness ability, it is no longer getting discarded. Instead, it is a spell on the stack. Once that spell resolves, it does go to the graveyard as usual, but this is a completely separate event from the card getting discarded.
This would not work as you are thinking.
The key is that Dina's ability does not do damage to your opponent. Although taking damage results in a loss of life, the effect "loses life" does not count as taking damage.
Lifelink only works when the source with lifelink actually deals damage:
702.15b Damage dealt by a source with lifelink causes that ...