TL;DR: A used board that supports VT-d if you can find one and are willing to buy used. If not, whatever board is cheapest here and meets your I/O requirements (currently an ASRock H81M-HDS R2.0 looks like the best option.)
I'm assuming you've checked and your board is out of warranty and that RMAing it isn't an option. If not, I'd recommend you go that route (though from anecdotal experience, Gigabyte is particularly poor with RMAs.)
First, let's take a look at the various chipsets for the LGA 1150 socket.
You'll notice that as we go down the list, the boards essentially get better.
Straight away, we can ignore some things: since you've got a Haswell CPU with no plans to upgrade, we don't care about having a _97 series board for compatibility purposes. Again, since you have no M.2 devices with no plans to upgrade, we can ignore the _97 series on that front.
Since your CPU doesn't support overclocking, and you have no plans to upgrade, getting a Z series board is unnecessary for overclocking. As such, no motherboard will offer a performance boost, assuming it supports your RAM speed, which–based on the tier of the rest of the system–I'm assuming isn't particularly high, and is likely supported by all motherboards.
After that, the main difference is mostly I/O. Your requirements are quite low, and any board at all will fit these requirements (at least 2 SATA 3 ports and one RAM slot.) You mentioned "max 6 USB ports", but didn't clarify if you needed USB 3 on those ports, nor how many were provided by the case. I'll assume for now that 2 USB 3 ports, 4 USB 2 ports, and one USB 2 header is enough, which puts almost any motherboard on the list. I'm also going to assume you want an HDMI port on the motherboard, and that you'd prefer not to get a (relatively inexpensive) passive DVI to HDMI adapter. You also mentioned wireless keyboard/mouse, but since your current board doesn't have built-in Bluetooth, I'll assume you already have some solution.
However, your CPU does support one interesting technology: VT-d. Basically, it allows you to directly pass through I/O devices (like GPUs) to virtual machines with no* performance degradation. Though it's not for everyone, as a developer, this can be quite useful. Though Wikipedia lists only Q87 boards as supporting VT-d, this is incorrect. For example, the ASRock H87M Pro4 supports VT-d, as does the B85 Pro4. However, so far, I've been unable to find an H81 board that supports VT-d.
LGA 1150 boards are new enough that there's enough stock lying around that getting a new board won't cost you too much extra while providing you with a warranty and other nice benefits... if you're willing to use a Micro ATX H81 board. This prevents you from using VT-d, and will limit both your I/O and expansion options. Buying other boards new is significantly more costly, with the cheapest not-H81 board currently being a ECS B85H3-M at $63.32 (though it appears to lack VT-d, defeating the main point of getting such a board), followed by the Biostar TB85 (again, also apparently lacking VT-d), and the Asus B85M-G R2.0 at $96.59, well above what the used market is at (roughly $40), while having poor to mediocre I/O selection.
Of course, buying used comes with its downsides. If you want the best value, it will also take some searching–Craigslist and eBay auctions can be a significant hassle (I couldn't find any eBay Buy It Now offers that fit your requirements and were decent value.) If you're not comfortable with that, I'd suggest saving the money and just getting a cheap H81 board, unless you have an immediate use case for VT-d. Those used boards will still be around.
As always, make sure your case will fit the board too!