Lilian described Doris as her "best friend" and said she felt lucky to "have had her by my side for 96 years".
Doris died on January 5 after she and her 96-year-old twin sister were admitted to hospital, having both tested positive for coronavirus.
Writing on their Facebook page, Mrs Cox described Covid as the "biggest fight" of her life but said she has been "feeling stronger" each day.
Lilian and Doris, known affectionately as the Tipton Twins, rose to fame in 2019 and were the UK's oldest identical twins.
Lilian is now raising money for Beacon Centre for the Blind in her sister's memory - a charity which was close to Doris' heart - and her appeal has now amassed more than £3,300.
She said: "I would like to say a huge thank you to everyone for their support and lovely messages this week.
"And a special thank you to Piers [Morgan] and GMB [Good Morning Britain] for my beautiful 'Get Well Soon' flowers.
"It has been a very sad week but I know Doris would want us all to keep smiling and remember all the joy she has brought to our lives.
"I feel so lucky to have had her by my side for 96 years. She wasn’t just my sister, she was my best friend.
"We created so many special memories which will stay with me and for that I’m forever grateful.
"Today is the first time I’ve wanted to get dressed since leaving hospital. Covid has definitely been the biggest fight of my life but each day I’m feeling stronger and I’m slowly starting to feel like me again.
"Thank you to everyone who has donated to the memory page I’ve set up.
"Doris would be so proud of this page and it’s nice to be able to raise money for a local charity close to her heart.
"Please stay safe everyone, stay home, save lives. Sending love and best wishes to each and everyone of you. Lil."
Doris' funeral will take place on Monday, February 15, where she will be laid to rest with her husband Ray, whom she lost 11 years ago after 65 years of marriage.
To donate to the Beacon Centre in memory of Doris, visit justgiving.com/fundraising/dorishobday.