Q:

What to do if a router doesn't seem to start up? 

It seems like a rare occasion since this isn't asked yet which surprised me a bit. 
Since this question is rather general, I will shrink the scope of this question for you a bit. 
My father bought a TP-Link N750 (TL-WDR4300) a few years ago.  Now the problem occurs that the router doesn't want to start up for some reason.  Funny fact is that the cabled ethernet part still works, but the WiFi part doesn't work, it does not want to reset, and the control panel (192.168.0.1) is not accessible.  So it seems this router can't start its wifi module and whatever controls the firmware on that thing. 
My question is now:

Is there anything I can do in the first place? 
Can I somehow still force a hard reset even if the reset buttons don't work? 
Is it worth the risk/effort? 

A:

These are actually  pretty decent home routers, and if you are lucky enough to have a v1.7 you can get dd-wrt on it.  The unit may just be defective, but if its just corrupted firmware these are well worth reviving. 
Solution:
1- Download 12-22-2014-r25697 factory-to-ddwrt.bin
2- rename firmware to wdr4300v1_tp_recovery.bin
3- Download tftpd32 version 4.50 (if you have firewall software, grant all permissions to tftpd32 and especially port 69.) 
4- set your PC/Laptop IP address statically to 192.168.0.66, Subnet Mask 255.255.255.0 (the router will be 192.168.0.86 and won't ping back if you try)
5- Connect your computer to the LAN port 1 on the router (WAN port won't work for this)
6- Open tftpd32, in TFTP Server select server interface 192.168.0.66
7- Hold down the WPS/Reset button on the router. 
8- Power on the router. 
9- After approximately 7 seconds release the reset button. 
10- In tftpd32=>tftp server, in the current  directory, browse and select folder with renamed firmware (in step 2)
11- The router will now download the firmware from your server and upgrade. (! ~Be patient and do not interrupt this process~!)  seriously be patient here or you can totally brick the unit. 
After the reboot it should be giving out dhcp leases again, dont forget to re-enable auto dhcp on your computer. 
Good luck to you sir! 
This info came from the dd-wrt forums.