The good news is you will be employable even if you can’t finish.
If they allow you to continue but require a 3.0 to graduate - even if you continue, it may be for not.
You should have an advisor you can speak with or make an appointment with the Dean’s office.m if you want to push.
ADHD self reports may be insufficient . We already know about the problems with self-report data 
Insufficient for what? And I wasn’t suggesting the OP has ADHD. I was just suggesting basically why keep banging your head against the wall. Better to get at the root of the problem. Whatever that is. It could be many things but whatever it is, it seems to be continuing.
I have no idea if accommodations are available in a master’s program if they should return, and that isn’t relevant until an evaluation of some sort is done, which the OP says they have been intentionally avoiding because they don’t want to know.
This student says he did well in his electrical engineering program. It’s the masters in mechanical engineering that is where he has had some trouble.
@TStixMo electrical engineering can be a very interesting and varied career. Have you explored the various types of jobs EEs do? If you don’t end up returning to your masters program, please consider looking at a bunch if the different options an EE can have. You might find this satisfying.
“Insufficient for what?”
Completing a Self report checklist is to most providers insufficient for a diagnosis of ADHD. It’s way too subject to response bias and not even at times face validity at its best. It may be a preliminary screening tool or part of a more comprehensive assessment. And there was no indication that this poster has ADHD. Sounds like the OP had family stressors contributing to their difficulties.
The OP has not indicated any ADHD symptoms (and did well in an undergrad EE program).
This is what my response was to. But as others have already stated, the OP has not indicated that this is a concern.
I wrote “I would be more concerned about why you have not been able to achieve the desired grades. Have you considered an evaluation for a learning challenge, ADHD, depression, whatever…..If those aren’t relevant, is a different school a fit? A different academic focus? Would working for awhile clarify things?”
Then I wrote: “ If you think an evaluation will be helpful, please pursue it. If you think it is ADHD primary care can often diagnose: it is just a questionnaire that you can find online. I am not into diagnosing online, obviously, so that is hypothetical.
I have been around long enough not to mention something like ADHD as anything but a hypothetical example! The OP indicated that they have been avoiding an evaluation but did not specify what it was. I was addressing that.
I took my kid to a lengthy neuropsych. as well as a prominent ADHD clinic at a major urban hospital and in both cases they used the questionnaire. I was not suggesting self-reporting. But looking at it online can motivate someone who is avoiding evaluation to seek it.
Don’t see any reason to post more. Not a debate!
Let’s return focus on the OP. Ask them questions to probe if necessary, but don’t presume facts not stated
OP- since your thread is about writing an academic dismissal appeal letter, right now that is your focus. I don’t think you have had a chance to answer my previous question- what is your deadline? When do you have to submit the appeal? As most here have said, the letter should be short and to the point. State that You made an error/misunderstanding in an early assignment that carried a lot of weight, and then you were then dealing with family medical crisis that took a lot of of your time and attention. You want to address briefly and succinctly what the concerns were and how you plan to address them. I believe you mentioned that you had a faculty member that was willing to be supportive of you and yes, reaching out again to that person is important although as I stated earlier, it’s the holiday time and they may be limiting their time on their school emails.
It’s best to do first things first. If you have a deadline and must submit your appeal, get that done. If you want to share a draft of that here for feedback, that’s your choice too. But writing that appeal needs to be a priority right now. There may be time down the road to explore what other factors may have contributed to your change in your academic success.
If you have continued struggles academically that are unexplainable to you, and you do consider pursuing an evaluation, getting on someone’s calendar is likely going to take quite a while. As a career neuropsychologist, if you have any specific questions about that or the process, feel free to back channel me.
Thank you for your insightful replies.
The deadline is the 12th of January to submit the appeal. I’ve produced a draft of the letter based on all the helpful ideas in this thread and will attempt to reach out to my advisor first thing tomorrow. Like you said, it’s the holidays so I’m not sure if I’ll get a reply right away, but it’s probably best for me to get the ball rolling at this point.
Thank you for your kind post. Hoping that your faculty member will be responsive in a timely fashion for you. And, FWIW, I have 2 engineer s’s (one mechE, one ChemE). Both did use their engineering backgrounds/degrees in the first few years of their careers, but their careers have evolved in different, but very positive directions. The good news for you is that your pursuit of an advanced degree is optional. You are employable now, and a variety of opportunities can arise once you are in a position. Keep us posted!