The cold days continue—how have you been?How have you been? It’s still cold here.

This is great grammar but the wording feels a bit mechanical. This is what I would write in a letter.

SI’m sorry again for the late replyaking so long to get back to you.

Again, your grammar is perfect. This is a stylistic suggestion.

I'm writing this letter while I’m in a clinic, waiting for my appointment, seated in the clinic's waiting room.

Actually, I was involved inI had a minor car accident with a car.

Please don’t worry—: the injuries aren’t serious.!

Ever since the snow began tostarted pileing up, it’s been difficult to get to my assistant jobmake it to work.

What normally takes about forty minutes by bicyclon my bike can sometimes take nearlyalmost twice as longthat.

On days when it’s snowingWhen the snow gets really heavily, I can barely see ahead of min front of my face.

It’s a wonder I hadn’t gotten into a majorlmost surprising I hadn’t had an accident beforeuntil now.

There were manyI had a lot of close calls.

Theis accident happened early in the morning, on my way home from work.

The snow from the previous day had formed deep ruts, which had frozen over, and then fresh snow had fallen on top, making ithem hard to see where it was safe.

A car cominghit me from behind hit me, and I fell sideways with my bicyclke.

Fortunately, the car wasn’t going very fast, so I escaped with justjust have a bruised shoulder and a sprained ankle.

You can say “escaped” but that sort of language sounds like an action movie!

Don’t worry—Fortunately, my right hand is fine.

It was my first time riding in an ambulance.

If I ever write a scene involving one, I think I’ll be able to capture the atmosphere vividlywith one, at least it’ll be a vivid scene.

I may sound carefree saying that, but when IThat might sound glib, but I feel terrible thinking about how much I worried my parents, it pains me deeply.

Especially my mother—s. She rushed to the hospital as soon as she heard the news.

“Especially my mother,” is a sentence fragment, so it isn’t “proper” grammar, but that’s very common in both written and spoken English. It’s breaking rules, but it’s OK!

I was so sorry I couldn’t even find the words to say how sorry I was.

MEven more than the pain, what tormented me was the shame.

I added “even” because that intensifies the tone.

InAt times like these, I feel keenacutely how I’ve failed to become thea son my parents canould be proud of.

The word “keenly” sounds a little dry or academic. I don’t hear it very often in conversation. The only grammatical “mistake” I see—and it’s not really a mistake but just an area that sounds weird—is that we tend to say “at a time” or “at times.” We say “in a time” or “in times” for very long stretches, like time periods or eras of our lives.

Until my leg heals, mMy mother will be driving me to work until my leg heals.

It’s embarrassing, but my injured ankle is immobilizedn a cast, so I can’t ride my bike.

You can say “immobilized,” but I hear that word more often in descriptions of an emergency while the emergency is happening.

When I bowed my head and apologized for the trouble, she snapp I caused her, she yelled, “Even after it heals, I’llm going to keep driving you until the snow meltsis gone.

I couldn’t bear it if you died next timeon the road.”

Saying “next time” has a subtle implication that she fully expects that another similar accident will happen again, no matter what she does.

Living in this town, aA car really is essential for living in this town.