Dirk is part of the finale event for arc 2.
Where the temple chief decide to invite a foreign noble to come and kidnap Myne away to get rid of her, he also promise Delia that the noble will also take care of Dirk (despite how much she's come to trust Myne, Delia still trust the temple chief too)
But then it was revealed that underneath the adoption contract Delia saw was actually slavery magic contract i.e. they lied to her and her baby brother isn't going to live as noble but as a slave.
Myne fought against the noble long enough for the cavalry (mainly the lord himself) to arrive and apprehend them.
The noble and temple chief immediately go "execute that commoner! She attacked nobles!"
And the lord play UNO reverse card on them "Actually, she is my adopted daughter now, meaning YOU attacked THE lord's daughter"
(he gave her a magic tool and tell her if shit hit the fan to activate it, it basically give the lord a legitimate reason to push the attackers instead of them getting away with 'noble vs commoner' reasoning)
So temple chief is executed for that crime, the noble get imprisoned and interrogated (he is still a noble from a stronger territory)
In the aftermath, Myne manage to talk the lord into not executing Delia for her part in the conspiracy, and the lord demands Myne be the one to give her a fitting punishment.
Myne then have the sentence for Delia be that she is to be confined to the orphanage, never be allowed to leave it ever (since it's a place of her trauma + never being allowed to leave means she can't go out to do attendant jobs)
Delia could read between the line (it means she can stay and take care of Dirk) and accept her punishment.
Then jump into much later in the story: After a LOT happened, one of Myne's ideas is to give the children of the traitorous nobles (they were sent to orphanage instead of execute because Myne argued that the children did no wrong, and in fact one of them betrayed his own parents to warn her of the plot) the chance to rejoin noble society, expand slightly into giving anyone with enough mana the chance.
Basically, any children at the orphanage can choose to take tests and if they pass they'll be adopted under the lord's name and send to magic school. One of the criteria is of course having enough magic power (the other topics were, amusingly, something Myne already educate her orphans anyway like reading/writing/calculation)
Dirk volunteered and passed (of the 3-4 tier of nobles designate by mana pool, he's mid-tier)
iirc some of the ex-noble children were fine remaining in the orphanage to become blue priest/ess
The revelation that he is leaving the orphanage CRUSHED Delia, who wanted to remain together.
But Dirk manage to convince her in the end, saying that right now they're ok, but they can't be sure if future orphanage director (Myne is leaving the territory after getting one of her own....) will be as kind to Delia given her criminal record.
So Dirk will go to the magic school, gain status of a 'real' noble, and return to take the position of orphanage director, so that this time he can be the one to protect her ( I believe he can make her his attendant since the orphanage director room is technically part of the orphanage)