inadequate RN :(

It sounds like you are doing private duty nursing since you are doing shift work. It's not recommended for inexperienced new grads unless an extended orientation with skills lab training. You should not be trained by the parent, you should be demonstrating competency to a preceptor or clinical nurse supervisor. Parents can orient to the routine and care specifics.

You should not assess for ear infections therefore an otoscope is neither necessary or appropriate. (Aside from pain, fever, discharge, pulling at ears then you contact the patient's physician with your findings. The MD will decide whether your assessment is sufficient to call in an Rx or needs to see the patient in the office)

The home should have a thermometer or one provided by the agency. You provide your own stethoscope unless isolation issues then agency provides one. Agency provides gloves, PPE, hand sanitizer, soap, paperwork, care plan. DME supplies pulse oximiter, oxygen, BP cuff, nebulizer, diabetic testing supplies & sharps contained, nebulizer & respiratory care supplies, suction and any other items prescribed in the plan of care and signed by the physician and appropriate to the diagnosis that creates a need for skilled nursing.

You don't assess other than what is in the plan of care. I.E. You don't check pulse ox with a personal device ever. Only use home device provided by DME and should have oxygen, suction, nebulizer etc to provide treatment based on results. A child without a respiratory diagnosis will be T, P, R only

Don't beat yourself up. If the patient wasn't exhibiting any s/s of an ear infection, how were you to make that discovery? This stuff happens ALL the time in home health. I've had patients walking and talking and the next day, code. You can be as vigilant as possible but unforeseen occurrences can and will happen in home health just as in a hospital setting. Best of luck!