Second vent stack required?

[Recovering Perfectionist]

See attached picture. Should make everything clear!

I'm new to the forum, first post, but I have lurked a lot to get some of the best answers on this site above all others. This is going to be long winded so bear with me. I recently purchased a home that was a flip done by a father and son. That's the first problem. Most work was done very nicely but others such as electrical and plumbing you can tell was done by non plumbers and electricians. I'm not a contractor by profession but have always done my own work. I always research the current code to prior to starting a project.
The single story home is located in Florida and was built in 1948. A laundry room was added next to the kitchen. The stand pipe for the washer ties into the lateral for the kitchen sink. The original drain line was replaced in the kitchen but no one was fore thinking. The original line sits 24" above slab. Fine in 1948 but with today's deep sinks and garbage disposal it doesn't work. They installed a drain pump since the disposal sat below the lateral. Problem is it pushed water back out the stand pipe in the next room. I removed the drain pump and disposal, and due to the sink depth the p trap sits too low. No way around it, I need to lower the lateral to maybe 16" above slab.
Question is, do I need to add a second vent line after the washer and before the sink since it is in series? The washing machine stand pipe is 6'4" from the original vent. A bit over the 5' I believe code allows.? They also used 1 1/2" pipe so that will have to go since the stand pipe needs to be 2". And the stand pipe is only 11" Tall! That will need to be a minimum of 18". Should work once I drop the lateral down to 16" off the floor.

Let me know all your thoughts.

PS, the vent and drain will need to be snaked. Once the pipe fills it takes 10 minutes for the sink to drain. Has to be obstructed! Also this is a second vent not the main stack. All other drains in the home work fine and are based off the main stack.

Thanks in advance!