Is the battery original? Most batteries don’t last more then 5-7 years before having serious degredation issues. Dell should still sell OEM packs for your laptop.
If you find the computer needs expensive repairs that are not worth doing, the tradition of long lasting batteries sort of lives on with the Dell Latitudes as I’ve had packs work after 7 years like it’s nothing (but I also wouldn’t trust it not to suddenly take a hit) while others only have 50% of their life left, but it’s a far cry from the endurance of the Sony batteries - which ran like they were new 5-7 years down the line. I thoroughly believe the vendors caught on the Sony packs didn’t die in 1-2 years, so the newer packs are intentionally nowhere near as long lasting unless you spend more upfront like the grey label (3Y warranty extended life) Latitude batteries OR they don’t offer long life packs to keep their revenue up.
Rather then purchase a battery, remove it and see if the system boots properly without it - no need to sink money into a battery until you *know* it’s not a deeper and potentially more expensive issue.
Since the battery didn’t do the trick, is the hard drive original or replaced? If the drive has never been replaced, it may be failing due to high hours and excessive wear. Check the health with a tool like Parted Magic and see if there are any red flags like excessive reallocated sectors or other serious SMART errors. It’s new enough to be from when Dell went full swing and started to more towards Seagate exclusively, so I wouldn’t be shocked if the drive has issues. It’s actually very common for post 7.01/7200.11 drives to fail at the rate they do compared to the older Seagate drives of the past.
The WD Blue option (less common, but it shows up) isn’t a great drive either. The WD Blue drives Dell loves to use aren’t as long lasting as the WD Black series and tend to be just as failure prone as the Seagates, but you usually don’t have as many issues with the Blue drives compared to the Seagate systems.
Drive replacement on the N5110 (15” version) requires a total strip down, so that model is horrible compared to the N7110. On the N7110, the motherboard doesn’t need to come out - but you still need to remove the RAM door, a lot of screws, the keyboard and palmrest :P.
If the drive is bad, most of them are now Fixed III or II/III Premium but you’re in the clear since you have a new enough machine to work with Fixed SATA III SSDs. You can buy a cheap DRAM-less SSD like the PNY CS900 or spend a lot on a Samsung autosense drive at will since it really doesn’t matter in your case. If the drive doesn’t fix it, it may be time to throw the towel in and get a new to you Haswell machine like the older Latitude E6X40 series and a HDD optical drive adapter if you want a second drive - the E6X40 series still uses mSATA despite being beyond dead by then and you have to pick between an SSD (mSATA) or WWAN. Buy it on eBay so you can get the ownership transferred - although Dell can be quite annoying with eBay systems since nobody knows the previous owner and you SOMETIMES need to back them into a corner in order to get a proof of purchase request.