I know I'm resurrecting the dead here, but here is an "half-answer" for anyone who stumbles across this question. (I know you've already sorted your issue, and no one wants to reboot their servers if they can help it!)
You can do this from OpenManage for ESX, but...
...as it does require a reboot of ESX after install, at least if you pre-install it, and if you have some unexpected downtime, when the server comes back up you can have all the access you want, as well as change the iDrac username/password.
Here are instructions from Dell, and supports ESX 5.5 to 6.5 - steps 1-6 tell you all you need to know to install OpenManage on ESX, just ignore the last step.
I make it a habit now to install OpenManage on all servers immediately after installing ESX as well as setting up the iDrac, just in case!
(I'm not going to re-post the entire page contents - suffice to say if anyone comments on this that the link is broken, I'll update the link)