Delayed and detained
The influence roll on the Detained card is a Local Action, meaning any investigator on the space, yourself included, can roll to perform the action.  Normally, you would be able to roll on it to try to remove the Restriction.  However, being Delayed adds an additional restriction indicating that you cannot perform any actions.

I believe the question is based around if the act of standing up from being Delayed is an Action.  As the rule book states:


  A Delayed investigator cannot perform actions. 


The only logical way for an investigator to cease detainment is for the act of standing up to not count as an action.

The net result for your round would be:


Any investigators before you, who are also on your space, can roll as a local action to try to discard your Detained card
On your turn, regardless of Detained status, you stand up and perform no other actions due to being Delayed
Any investigators after you, who are also on your space, can roll as a local action to try to discard your Detained card
During your encounter phase, if the Detained was not removed, flip the card and encounter the back