Those sentences sound unnecessarily convoluted to me, and "would" doesn't seem to be a good choice.
"would" in such real situations as in your examples is only used when you're not sure about something ("I would say it is ..."), or to soften the tone when making a request ("I would like to ask you to ..."). In my opinion, your examples don't fall under these categories and should use the more clear and definite "will".
I think most people will say those in shorter phrases in the active voice and not using "would".
1. As I mentioned, I will update you (on [the subject]) by email after this chat session.
2. Rest assured this will be looked into and will get fixed. (even better: Rest assured we will look into this and fix it)