Weird, CPUs rarely break. Anyway, a 760K is pretty weak so an upgrade may be in order regardless.
Current Intel hardware uses DDR4, so you'd have to get a new CPU, motherboard, and RAM but could reuse the rest. This would give you an i5/i7 6xxx/7xxx and you'd want to use something like the Z270 or H270 chipset.
Older Intel hardware uses DDR3, so you could reuse the RAM. It sounds like you have 16GB, which is relatively pricey. Even this "older" hardware is still an upgrade from your current motherboard, though, so going with an Intel i5-46xx(K) isn't a bad idea. Though newer games can benefit from the i7 CPUs, but OTOH with a GTX960 it probably doesn't matter. Here you'd be looking at Z97 or H97 chipset.
The Z chipsets are for the CPUs that end in K as the Z chipsets allow overclocking and the K CPUs have unlocked multipliers. Non-K CPUs can't overclock, but you can still use them in Z chipset based motherboards. The H chipsets do not have overclocking, but you can use a K CPU in them - it's just stuck at stock speed so it's generally a waste to pair these together. The -K CPUs, in retail packaging, do not come with a heatsink but the non-K CPUs do come with a stock heatsink (it's not great but it does the job, since it's not overclockable anyways you can't void your warranty by using it).
Aside, but your 1TB WD Black isn't memory, it's storage. Memory would be your RAM.
Figure out your budget first, and assuming you'd keep your GPU, Storage, PSU, and case you can figure out what to do. Basically if you have $300 then you can get an older i5 4xxx and H/Z97 board. If you have $400 you can get an i5 6xxx/7xxx, an H/Z270 board, and DDR4 RAM *or* an i7 4xxx, and a H/Z97 board. If you have $500 you could get an i7 6xxx/7xxx, an H/Z270, and DDR4 RAM.
If you can do the current generation hardware, you'll get a better CPU (slightly better raw performance, but improved IPC and efficiency). There are some worthwhile changes from the H/Z97 to H/Z270 boards. One is M.2 PCIe 3.0 x4 which is the 32Gbps SSD interface. H/Z97 only had M.2 with PCIe 2.0 x2 or SATA-III, which offered either 10Gbps or 6Gbps respectively. The other is USB 3.1 Gen 2, the 10Gbps USB protocol. Another is more PCIe lanes from the chipset. And of course DDR4 tends to outperform DDR3. However, if you don't plan on using an NVMe SSD, don't plan on using any 10Gbps USB devices, and don't plan on having any other PCIe cards then none of these really matter to you. Even the memory difference is generally only noticeable when you benchmark memory, outside of that the real-world impact is hard to discern since you have different CPUs in the benchmarks and that's a much bigger effect than the speed, timing, and generation of the RAM modules.
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