How do I get a non-probate from state that I am the administrator of my own estate after wife died?

I am very sorry about your wife's passing. I understand it is frustrating to have to deal with the issue you mentioned, especially when you are also grieving for the death of your wife at the same time. But you should be able to get the records you seek, it will just take you opening an estate for her and being appointed as personal representative. 

To get your wife's medical records, usually that requires you to be the personal representative of her estate, not your own estate. You could never be administrator/personal representative for your own probate estate, because probate estates are opened only for people that are deceased. 

It sounds like you do not need to open up an estate for your wife to transfer any assets, as all assets you owned were jointly with you and your wife and/or you were names the beneficiary. However, it sounds like you do still need to open a probate estate in order to be appointed personal representative, so that you can obtain the records you need. That is not a problem, as people often need to open estates for this exact reason. There will be no assets to deal with, and the sole purpose will; be to get the medical records. 

Even without a will, as the surviving spouse you will have priority to server as personal representative of your wife's estate. To open the estate and get appointed as personal representative, you need to fire the necessary pleadings in the court in the county your wife resided when she passed away and give other interested parties (like other heirs, if there are any) notice depending on the circumstances. You can find the necessary pleadings (called "JDF" forms) online, as well as instructions on opening probate for your specific situation. If you have further questions on the specific pleadings to file, I will also advise you to consult with a probate attorney regarding your specific case and questions. Our firm regularly handles this type of case. including filling out and filing the necessary pleadings to open a probate estate and get a personal representative appointed. 

After you are appointed by the court as personal representative of your wife's estate, you should be able to submit a request to the medical provider for records, along with a certified copy of your Letters of Administration (which the Court will issue you once you are appointed). Some providers also require you to fill out an authorization form, as well as a death certificate. 

To second your question, about whether you need a will or a trust to provide for your daughter, the best state planning solution for you depends on your specific circumstances. I would recommend consulting with an estate planning attorney at a reputable firm. Our firm regularly handles estate planning for clients, and your initial consultation would be free. Generally, you would likely be looking at a will, trust, or a pour-over will/trust combination, and factoring in what assets you can leave outside of probate via non-probate transfer (beneficiary designations, POD, etc.). But there are other issues to consider as well, depending on the age of your daughter and your overall goals, such as the who you would want to serve as her guardian on the event of something happening to you, or who you would like to serve as your power of attorney, guardian, or conservator in certain circumstances. Those are issues that should be discussed in coming up with any comprehensive estate plan. However, to give you some peace of mind right now, the default (without a will/trust and without beneficiary designations) if you are not married and have only one child (your daughter) would be that your daughter receives all of your assets when you pass away. It just may require probate for her top receive those assets, depending on how they are titled.