The total number of patients seen in December, including walk-in centres - was up by 983 to 17,078 last month.
Chief executive at JPUH Christine Allen has said previously of the struggles faced by the hospital this winter: "The latest figures simply reflect the exceptional pressure that our hospital has faced since the New Year period".
The letter to the Prime Minister said: "We feel compelled to speak out in support of our hardworking and dedicated nursing, medical and allied health professional colleagues and for the very serious concerns we have for the safety of our patients".
Dr Clifford Mann, A&E national clinical adviser, said: "We need better bed occupancy levels during the winter".
The consultants draw on their own experiences - including patients dying prematurely, waiting times of 12 hours and more than 50 patients waiting for a hospital bed.
"I've heard an overwhelming number of reports from clinicians about how flu and respiratory illness, combined with other winter pressures and exceptional and sustained levels of demand, is changing the way they are treating patients arriving at A&E".
Nearly 15% of England's A&E patients waited more than four hours to be seen last month - the joint-worst figure on record - as senior doctors warn of patients dying in corridors.
Going on to add that the NHS is severely and chronically underfunded and has insufficient resources to deal with an ageing population.
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Ms Davidson said: "People are waiting too long in A&E departments because there are no beds for them on wards, and because many of those hospital beds are taken up by patients who are waiting for their social care arrangements".
In an unprecedented move, doctors told May that 120 patients a day are being managed in some hospital corridors, with "some dying prematurely".
University of Hospitals of North Midlands NHS Trust was among the worst for hospital trusts in the country for the percentage of patients seen within four hours - 71.5 per cent - over the month.
Questions have been asked in recent days about low flu immunisation uptake rates for NHS staff, with HPS again urging everyone who is eligible for vaccination to do so.
"The NHS today has launched their national flu programme and I would encourage people to act on the advice that the NHS is giving and also encourage NHS staff who haven't had the flu vaccine yet to have that vaccine".
The Royal College of Surgeons said it was "disappointing" that just 77.3 per cent of patients at major A&E departments were treated within four hours, worse than in December 2016 when 79.3 per cent of patients were seen.
A spokesman added: "It further demonstrates why it has been necessary to cancel patients' non-urgent procedures until the end of January".
Medical director Andrew Murray said: "We are sorry that a number of patients have experienced longer waits during this busy period and we are doing everything possible to reduce delays".
It comes as new figures revealed the number of flu cases in Scotland had more than doubled over the past week, leaving them four times higher than the same time a year ago.