Battery ran offut

In the sense that there is no power left, we say a battery has "run out"

Early in the morning yesterday I was ready, while preparing to drive to work , I found that the lights of my car were still on.

The battery totally went offut.

Continued from above, although "go out" (or "went out" here) has an alternative meaning of shutting down

But I insisted on fixing it right now cause I do not have time and always work over time.

Anyway, in the end, I got a car repairman to help me charge my battery.

Early in the morning yesterdayYesterday morning as I was rleadyving to drive to work , I found thate lights of my car were still on.

Battery ran offut

Early in the morning yesterday I was ready to drive to work when I found that lights of my car were still on.

"when" is needed here to connect the two halves of the sentence :)

Then I realized I'd forgotten to turn the lights off and had left my car all night abeside the road.

"Had forgotten" sounds more natural here because it's "the past of the past" – it happened further in the past, before the story you're telling which is also in the past. "Aside" doesn't sound right here; "beside" is at least correct, but if you wanted to change it more, "by the side of" sounds more natural again.

The battery went totally went offdead.

With batteries, "went dead" is a really common way of describing this situation. I also moved "totally" to after the verb, where I think it sounds more natural.

They suggested to me that I should go to work first and fix it after workwards.

"to" is needed after "suggested" to show who is receiving the advice. "after work" is not wrong, but because you only just mentioned work in the previous part of the sentence, usually we wouldn't repeat the word again.

But I insisted on fixing it right now then because I do not haalways work ove rtime and always work owould not haver time later.

"insisted on" because it's past tense, and you need the preposition "on" to make this sentence correct. "cause" is perfectly fine in speech, but in writing it should be "because". I also rearranged the order of the last part, because it implies a cause and effect: you always work overtime, SO you wouldn't have time later. "Would" is used because at the time you said it, you'd have said, "I will not have time," and "would" is the past tense of "will".

Anyway, in the end, I got a car repairman to help me charge my battery.

Your sentence was already correct, really, but the commas make it sound more natural.

Battery ran offut

Typically we say "out" instead of "off". As in: the battery ran out of charge.

Early in the morning yesterday morning, I was ready to drive to work when I found that lights of my carmy car lights were still on.

Introductory phrases should have a "," (comma) after them. Early in the morning yesterday can be shortened to "Early yesterday morning"

"I was ready to drive to work" and "I found that my car lights were still on" are two complete clauses: meaning, they can be sentences by themselves. Therefore, you need to connect them with "and" or another conjunction. In this case, I've placed "when" to show a completed action (seeing the car lights on) happens in the middle of a continuous action (you were still on your way to work)

Then, I realized I had forgotten to turn the lights off and had left my car all night aside the road.

You could have also said "I realized that I forgot to" but had forgotten sounds better to me as a native speaker. Probably because "had forgotten" shows it happened in the past but is still affecting you when this story happened.

The battery totally went offdead.

We refer to batteries as being "dead" and not off.

They suggested to me that I should go to work first and fix it after work.

In this instance "They" is the subject of the sentence and "me" is the indirect object. Therefore, you would say "to me" (or: "to him" "to her" "to them" etc.)

But I insisted fixing it right now then because I do not have time and always work over time.

Cause is okay in formal speech, but not usually we write "because"

Anyway iIn the end, I got a car repairman to help me charge my battery.

Anyway is unneeded here in written form, but would be quite acceptable in casual speech.

Battery ran offut (or "Battery died")

Early in the morning yesterday I was ready to drive to work , but I found that lights ofn my car (or "Car's lights) were still on.

Then I realized I forgot to turn the lights off and left my car all night aon the side the road.

The battery totalcompletely went offut (or "The battery was completely dead").

They suggested me that I should go to work first and fix it after workwards.

But I insisted on fixing it right now because I do not have time and always work over time.

Anyways, in the end I got a car repairman to help me charge my battery.