I’d cut it down even more- to let them see what they need to and show your maturity and ability to focus. You really cant show a new sense of focus if you wander through all those details. It does seem scattered.
“I received your email regarding academic dismissal and would like to respond and request reconsideration. I have been making efforts to resolve this and will include brief details of the particular challenges I had been dealing with.”
I hope you have revised your approach and didn’t delay until you go the letter.
“As of (date,) I’ve been taking the following steps.” Then the bullets, but starting with the most effective (to turn around academics. Not some vague maybes or career services or your parents, etc.) Most important steps, whatever they may be. Eg,
• Working with my advisor to create a plan to get back on track
• Going to office hours to improve my understanding of the material
• Adding more study sessions and workshops
• Dedicated weekend time to review lectures, test my understanding, read ahead and get started on papers
• Prioritized academics and dropped several activities until my performance is satisfactory
Hope you get the idea. Skip the vague "promises"and show the action you took. None of us can entirely redo this for you. They don’t need to know which classes met which days.
“With the 14 hours per week of internship at dental office as well as Organic Chemistry, Physics, Calculus and Nutrition, I was off to a good start of the quarter. I spent time at library after work to review morning lectures and get ahead on readings, with nothing else on my mind.”
Instead of asking for sympathy, simply state the challenges.
“On October 12th, 2015, my grandmother passed away after a long illness. In that same week, I missed Physics 142 mid-term exam for the viewing and funeral, returning October 19th. Though I was determined to make up this exam, my professor does not offer makeups, no matter the circumstance. This resulted as a drop score (0%)” I hope you did speak with the prof, about this, that would show you tried.
Etc.
My version may be too dry, too abbreviated, and I probably shouldn’t have offered this revision, but I wanted you to see the point we’re all making. This is serious. Edit, to stay on message. Good luck.