Hey J. -
It sounds like you are are in need of a few good mom friends. You are in the right place online! Hopefully you have older women around you who can help you and advise you. The "seasoned moms" in my life are irreplaceable to me... I have learned so much from them.
Being a Mom is exhausting, thats for sure. Three year olds ESPECIALLY are draining because they are still quite needy and dependent... but are also learning to be independent - not to mention the constant chatter! What I'm trying to say is that I understand how tiring it is. I am asleep before the news comes on at night. Sometimes I think I could go to bed at 7 and sleep through the entire night if left alone! This will all pass.
I am going to tell you the truth. To me, it sounds like you already know what the problems are. You said you knew she was behind in speech, that she watches too much TV and that you are too tired to be very consistent with teaching her letters, numbers and reading to her.
These are skills that are going to be required of her in everyday life, as well as preschool. Teaching her letters, numbers, how to clean up, how to tie her shoes, how to change clothes, how to go potty, how to put dirty clothes away.... how to speak politely and how to use the proper words for things.... how to sit still for a book, how to use silverware and how to wash her hands the right way.... this is your job. If you don't teach her these things, she will be seriously lacking in her foundations for learning.
What I mean is... she won't have much to build on. This is quite a vicious domino effect. What happens eventually is she learns to cut corners and be lazy. She won't wash her hands. She won't be very polite. She won't speak well. She will inevitably take longer to learn her alphabet... therefore she will inevitably need more time to learn to read... which can lead to her hating reading... which in turn will teach her not to study. Do you see the ripple effect? I hate to sound dramatic, but its important that you understand the bigger picture.
My solution? Take a multivitamin with breakfast and also a B-6 supplement (it will give you more energy). Eat more proteins and less sugars. Put post-it notes up around common areas in your home that you will see with encouraging phrases on them if you have to - or little reminders - anything to tell yourself to go and do something with her.
Schedule your TV's "On" time and "Off" time. For example - my TV is "On" from 6:30 am - 8:30 am. My husband and I have breakfast and watch the news and then cartoons are allowed after that until 8:30 or after I have finished cleaning the kitchen up from breakfast. ITs then "Off" until 3 pm. At 3 pm, I turn it on for the boys for one hour and they may watch a movie or they may watch 2 cartoon segments, their choice. At 4 pm I watch Oprah and start cooking dinner and the boys go outside to play. When Oprah goes off, so does the TV and I finish dinner. We eat. We hang out and play. Boys go to bat hand bed.
Sounds rigid - I know. But I also know that they aren't spending too much time in front of it. They are learning by default when the TV is off - they have to talk to each other and to me, and play games and color and paint or clean up their rooms, not to mention doing school (we homeschool.)
Write yourself out a family schedule. Stick to it. What will you and your family be doing from Wake to Sleep each day? Try hour by hour. Then take the day in segments. Schedule yourself for a nap. Don't over-do it, or you will give up. But change something, because if you do not - all of your worries and fears will be real. She WILL be slower than other kids. It WILL affect her learning. You are RIGHT to feel badly about it but its okay because you can change it! And when you do, you will be shocked at how much she catches on!