If your phone won't turn on or charge and was previously involved in water damage, you are out of luck. Your phone is most likely not repairable and possibly not worth the cost if it was repairable. Neither would it pass for warranty repair.
For those of you who just recently dropped the phone, and after that it started behaving this way ... then you are likely in need of a new LCD screen, or best case scenario you might open the phone and find a loose connector. (But before resorting to looking for repairs, still try the below suggestions).
For everyone else who just randomly got struck with this situation, try these suggestions:
1.) First in order to rule out screen damage, hold your Power button for 10 seconds. Wait 60 seconds. Turn the volume button down then up -- you're looking to try and get the phone to make a sound. (Repeat this process twice -- in case the starting state of the phone was ON rather than OFF). If it does make a sound, your screen needs to be replaced. Find a replacement LCD online for $40 and fix it yourself or look to go to a shop to fix it. If it doesn't make sound, then continue down the list.
2.) Try a new charging cable and or a different wall outlet. Leave it 'charging' with the new charger or outlet at least 15 minutes. Then, disconnect the charger for 10 seconds, and then reconnect and let it charge for another minute. Hold the Power button for 10 seconds.
3.) Connect your Nexus 5 to a PC or Laptop using a reliable charging cable
a.) Check to see if your PC at least recognizes the Nexus 5 (a message will popup in the bottom right corner -- Windows). If the phone is working (but just stuck in this weird state) the computer will detect the phone nearly right away. If it does recognize the phone, hold the power button for 10 seconds while it is charging from the computer.
b.) If it doesn't detect it right away, then leave it plugged into the PC for at least 15 minutes. Disconnect the charger and then reconnect it 10 seconds later. Let the phone sit charging for another minute. Hold the power button for 10 seconds while it is charging from the computer.
4.) If none of the above suggestions have worked yet, then its time to turn your phone into an old school video game controller and start punchin' in some 'cheat codes'. Since no one can come to decision on which pattern works for which device model, let's try each pattern. While the phone is charging;
a.) Press and hold, simultaneously, the Power button + the Volume Up + Down button for 10 seconds.
b.) Press and hold, simultaneously, the Power button + the Volume Up button for 10 seconds.
c.) Press and hold, simultaneously, the Power button + the Volume Down button for 10 seconds.
5.) If you got to this point and still haven't solved your problem ... you're probably screwed. There is one last ditch effort however; leave your phone charging in a reliable outlet, using a reliable charger for at least an hour. Disconnect the charger for 10 seconds, and then reconnect and let it charge for another minute. Hold the Power button for 10 seconds.
If everything fails, then the only logical reasons (from a technical standpoint) would be that either your battery is completely done, the charging port flex cable is damaged, or the motherboard has to be replaced. To figure out which is the culprit you will need to take apart the phone and use a volt meter to measure and see if the battery is holding any charge, or if the charging port flex cable is passing current. The battery or charging port flex cable is about $15 each while the motherboard is about $80. If it is the motherboard, the phone is 'totaled' and not worth the repair.
Note: If you are not good with small electronics, then do not attempt repairing anything yourself. But if you are brave, have still hands, and good with being neat and organized ( ... lots of small screws and parts ... ) then go on YouTube and watch a full deconstruction and reconstruction tutorial.
Hope I helped.