There was no romaji in the transcript so the translation is my own:
Mazu, o-haka o kirei ni o-souji shite agete kudasai.
First, please clean the grave.
Having "kirei" here sounds like it's saying the same thing twice. Kirei means "clean" and "souji shite" means "to clean". I could understand "o-haka o kirei ni natte agete kudasai" ("please make the grave clean") or "o-haka o souji shite agete kudasai" (please clean the grave) but "o-haka o kirei ni o-souji shite agete kudasai" ("Clean [in the direction of] clean the grave") is bizarre.
Sorekara o-hana to o-senkou o agete agetara ii deshou.
After that, you should offer flowers and incense sticks.
Again, saying the same thing twice. Wouldn't "sorekara o-hana to o-senkou o agetara ii deshou" be fine? Why is "agete" here?
Watashi nani mo oyakoukou dekinakute gomen ne.
I'm sorry I couldn't be a good daughter.
"I what too/among other things good daughter couldn't be, sorry." What is the point of "nani" here? My best guess is, "I couldn't be a good daughter [and also there were other things I couldn't do]" but I don't know how "nani" becomes "other things".
Yatto ichininmae ni natte, korekara da to omotte kaette kita no ni...
I've finally grown up, and I thought I could make up for it and come back, but...
This looks like two thoughts seperated by "to omotte".
Yatto ichininmae ni natte, "korekara da" to omotte "kaette kita" no ni...
I've finally grown up, "Then [I'll do it]" I thought, "I'll come home..."
Without "kaette kita" being a thought the character had in the past, the word order here becomes deeply confusing.
Otousan to okaasan ga jiko ni atta koto mo shirazu ni...
I didn't even know that you were in an accident...
I think "atta" here comes from "aru" like "an accident existed/there was an accident" but I'm struggling with the particle "ni". It's the direct object of the verb, isn't it? So why not "o"?
Demo otousan to okaasan wa maitoshi koko e kite wa Satoko-chan no koto o jiman ni hanashite kurete ita yo.
However, your father and mother came here every year and talked about you so proudly.
"Wa" is used twice in this sentence, I don't know why it is used in the second instance. Something like "toki" would seem more fitting rather than setting a topic ("parents") and the immediately changing the topic ("parents coming to the place every year").