Doris, who celebrated her 96th birthday in July, died earlier this month after contracting coronavirus.
She died on Tuesday, January 5, after a short stay at Sandwell General Hospital.
The family announced the news on the Lil and Doris Facebook page this morning.
Doris's funeral will take place on Monday, February 15, and she will be laid to rest with her husband Ray, who she lost 11 years ago after 65 years of marriage.
The family have asked for no cards or flowers, but instead donations to be made to a charity close to Doris's heart – the Beacon Centre for the Blind. The family said the charity helped Doris look after Ray for over 20 years when he lost his eyesight suddenly during their long marriage.
Doris's sister, Lilian Cox, was also rushed to hospital and spent two weeks on oxygen, steroids, vitamins and antibiotics. She was discharged from hospital on Monday and is recovering at home.
RIP Doris Hobday, 96.— Piers Morgan (@piersmorgan) January 20, 2021
One of Britain’s two oldest twins & a delightful lady.
She died from coronavirus.
Her sister Lil got covid too but survived and is now home from hospital. My deepest condolences to her on the loss of her beloved sister. 🙏 pic.twitter.com/VwRy0XUsdU
In a statement, the family said: "We apologise for not updating you all but we have experienced a terrible few weeks – shortly after our last post Doris and Lils health deteriorated very quickly.
"We are devastated to announce that Doris didn’t make it. She passed away on Tuesday, January 5, after a short stay in hospital. She was still running fast but unfortunately had some hurdles put in her way and she wasn’t strong enough to fight Covid.
"We would like to thank the doctors and nurses at Sandwell Hospital for looking after her and allowing her great niece Kerry into the ward, so she could spend the last few hours with Doris at her bedside where passed away peacefully.
"On the same day Lil was also rushed into hospital and had the fight of her life for two weeks on a mixture of oxygen, steroids, vitamins and antibiotics to keep her alive. Lil has fought so hard and I’m pleased to inform everyone that she was discharged from hospital on Monday, she was only told then of Doris’s death once she was strong enough to take the news, she is now being comforted by family and staying with her daughter Vivien while she fully regains her strength."
It added: "We are aware they are both 96 years old and we have been so lucky to have them in our lives for such a long time but they were both still going strong before this virus got them and both were determined to live until 100, they had so much to look forward to – it’s just so cruel that Covid has stopped Doris like this.
"Our thoughts go out to all the families who have experienced a loss through Covid and we urge people to take this seriously, Doris was only a few weeks away from being safe.
"With her vaccine letter arriving two days after her death – if you are offered the vaccine please take it, do not refuse it. Doris didn’t get this choice."
To donate to the Beacon Centre in memory of Doris, visit justgiving.com/fundraising/dorishobday