Struggling to snatch a New Grad Job (RN)

I've been applying left and right at hospitals and LTC facilities within the Wheaton and Chicago area and I haven't heard anything back from them, just rejection letters.

I used to have a job at a nursing home but they let me go after the training period ended (1 week), stating that they needed somebody who could handle all 40 residents on the night shift by oneself (all of them had very high acuity since they are across the street from a hospital). I did everything right, but was only able to handle about 25 or so at the end of the week. As much as I loved working as a nurse, I feel really bad just losing my first RN job at the nursing home when I did everything right. The only thing the Director of Nursing said was to find a job in the hospital where the training session is longer, or find another LTC facility that is slower paced.

So now I'm back to square one, searching for a Registered Nurse position anywhere within IL, preferably near Chicago or 30-50 minutes from Wheaton. I'll take on any RN job as long as I can prove that I'm very capable as a new grad RN.