Dear (Mr./Mrs./Ms.) Smith, 

I am sorry to inform you that I will not be able to start my job next week, though I promised you I would.

I ran into a problem. My mom needs my help. She recently told me that she will be having eye surgery this Friday. She hoped that I would be able to look after her, because my dad is on a business trip and won't be home until the following Friday. Therefore, I need to stay at least one week until my dad returns. My mom will not be able to see during this period of time, so it will be very inconvenient if no one is able to take care of her.

I apologize again for having to start my job late. Because of this emergency situation, I will delay one week and start work on July 3rd. If you can allow me this time, I will be very grateful.

Thank you for your understanding.

Yours sincerely,

Maggie

You are due to start a new job next week but you will not be able to because you have some problems.

Dear (Mr./Mrs.) Smith,

I am sorry to inform you that I will not be able to start my job next week. Though, I promised that I would be able to start.

I have run into a problem. My mom will have an eye surgery this Friday. She asked me to look after her because my dad is on a business trip untill next Friday. Therefore, I need to stay at least one week untill my dad comes back. My mom will not be able see during this period of time, so it is very urgent that I am the one to take care of her.

I apologize once more for being late to start my job. Because of this emergency situation I will delay one week and come to work on the third of July. If you allow me this time, I will be very grateful.

Thanks for your understanding.

Yours sincerely,

Maggie (please use full name)

The first correction works very well. Please remember their advice.

I corrected your letter in such a way that sounds very natural in English and also has a level of professionalism. Remeber that when writing to a boss or a co-worker you what to use very formal writing. Feel free to search the internet for examples of business letters and formats. Format is the key to sounding professional! When I was a intern I had to learn this lesson the hard way. I would get in trouble a lot because I wrote so informally in my correspondance.

You are due to start a new job next week but you will not be able to because you have some problems.

Dear Mr/Mrs/Ms (use as appropriate) Smith,

I am sorry to inform you that I will not be able to start my job next week. Though, I promised that before.

I ran into a problem, it is that  that my mom needs me urgently. She told me that she will have an eye surgery this Friday. She hoped needs me to look after her because my dad who wasn’t will not be at home as he has had a business trip till next Friday. Therefore, I need to stay at home for at least one week till my dad come back. My mom will not be able to can’t see anything during this period of time, so it is will be very inconvenient if no one is able to take care of her.

I apologize for being late to start my job again. Because of this emergency situation I will delay one week and come to work on the third of July. If you can allow me this time, I will be feel very grateful.

Thanks for your understanding.

Yours sincerely,

Maggie

Hi, I made some corrections hope they are useful.

I think when expressing a problem of this sort you need to emphasise the seriousness of the problem so that your boss will agree to your request. So that is why you can use "urgently". As this makes the problem seem more serious and immediate.

We usually say "I will be very grateful", even though "I will feel very grateful" is fine grammaticaly and expresses what we feel.