Under the DMCA, YouTube is required to reinstate the content within 10-14 days following receipt of a valid counter notice (unless the rights-holder files an action in court before then).

All your counter notice needs to include are the following:

(i) a physical or electronic signature of the user;

(ii) identification of the material that has been removed or to which access has been disabled and the location at which the material appeared before it was removed or access to it was disabled;

(iii) a statement under penalty of perjury that the user has a good faith belief that the material was removed or disabled as a result of mistake or misidentification of the material to be removed or disabled;

(iv) the user’s name, address, and telephone number, and a statement that the subscriber consents to the jurisdiction of Federal District Court for the judicial district in which the address is located, or if the subscriber’s address is outside of the United States, for any judicial district in which the service provider may be found, and that the subscriber will accept service of process from the person who provided notification under subsection (c)(1)(C) or an agent of such person.

So, just follow YouTube’s process (or send the notice to their agent, which you can find in the Copyright Office’s directory) and in your notice include the exact information required, copying the statutory language verbatim for items (iii) and (iv) (e.g. for (iii), “I declare under penalty of perjury that I have a good faith belief that the material was removed or disabled as a result of mistake or misidentification of the material to be removed or disabled”). You don’t need to do anything more than that.

Here’s a decent resource on the basics of counter-notices: https://www.dmlp.org/legal-guide/responding-dmca-takedown-notice-targeting-your-content

Edit: and good luck and keep us posted!