First of all your daughter's teacher should not be irritated. In fact they should be helping you to figure out what your daughter's problem may be - if there is any. I don't know what month your daughter was born, but there can be a big difference at this age between someone who started school when they were almost 6 as compared to someone who just made the cut off to enter Kindergarten. So if your daughter was born during the summer or fall, then she might be stuggling just because of her birth date. Education.com has some great articles about developmental stages of children that I suggest you reading.
Here are a few things to consider.
Have her hearing checked by a professional such as an ears, nose and throat doctor. The school can do a screening, but that is usually done by a volunteer. Check with your insurance company to see what is covered and which doctors are in your health care network.
Make an appointment with your child's teacher and ask that the ESE specialist is also there. Discuss the problems you have all observed and ask them what the next step is in order to start the testing process. This will most likely start with a screening to see if there is a problem. If they see that there is, then further testing will take place. Public schools offer services in all different types of therapy that your daughter might benefit from such as speech, occupational, and physical. The testing process can take a long time - sometimes even a year - so starting now is a good idea.
If you have the money, you can have your daughter privately tested to see if there are any developmental delays or learning problems. Some health insurances cover this, so check with them before paying out of pocket.
In the mean time, be sure to read to her daily. She needs to hear the language. Also spend some time talking to her about what you've read. This will encourage her to speak. She also needs to develop phonemic awareness - which is the sounds of letters. For different ideas, please read the an answer I gave another parent on the topic - http://www.education.com/question/childs-teacher-told-daughter-8/
I would also keep a journal of what is happening. Record meeting dates, test results, and summary of action plans. You are doing the right thing by starting early and then keeping an eye on things as your daughter gets older. With your help she will get be fine.
If you have a more specific question, please don't hesitate to contact me directly. Good luck!