If I try and fix my laptop power adapter can I damage my computer or endanger people?
Well the Amazon link doesn't help much because we can't see the jack itself, so post photographs of what you have. 

As for the question itself, it's a bit difficult to answer. In general, yes you can kill the computer and kill yourself and others if the failure is really spectacular and the circumstances are correct (say the PSU starts burning and the fire spreads). However if you're careful the worst thing that can happen is that the charger may die and you can trip the circuit breaker. 

In practice, if the jack is damaged there is little you can do. If the barrel is physically deformed, you can try to straighten it out. The plastic part itself is most likely injection-molded and it's very likely that you won't be able to open it without destroying it. You'll most likely have to find a plug of same dimensions (measure inner and outer diameter) and same number of connections. They are in general standard-made so you could obtain one from usual component distributors if you look hard enough. If it turns out that you have to replace the plug, be very careful when you remove it! I suggest that you do not simply cut it off. Instead carefully remove the outer-most layer of insulation near the jack and see if the wires inside are colored. If they are, cut the existing jack so that some part of the cable stays attached to it. Later on, use a multimeter on continuity (or resistance) to figure out which cable goes to which pin. In the rare case that they are of the same color, you'll have to first cut one of the inner cables, use multimeter on it to see where it's connected and then mark it. Repeat procedure for all wires inside.

As for potential dangers, the two biggest (which could later on cause chain reaction) would be shorting the wires inside the jack and extremely bad soldering of the wires to the new jack. In general the power supply should have short-circuit protection, overheating protection and so on, but it's of course best not to rely on them. I can't say more about this than to be very careful what you do and plan each step you're going to do ahead of doing it and know what to do if something interrupts you during the step.

The short between wires can be detected by using a multimeter and probing contacts to see if there's continuity between them. t may be difficult to do that on the barrel of the jack, so be sure to think in advance how to do it on the wire contact side of the jack. The bad soldering will show up as either bad connection (which can get better or worse as the cable is moved) or as abnormally high resistance. To check the resistance, short the probes of your multimeter and measure the resistance between them. Then measure resistance between barrel and wire contact side of the jack. It shouldn't be higher than the resistance of the probes (but be sure to connect the probes to the jack properly). As for bad connection, try moving the cable and keep checking resistance while you move the cable. It shouldn't change.

Finally make sure that you power on the power supply in a safe area after the intervention (and be sure to connect it to power with no computer attached first). In a reasonably bad scenario, the power supply may have an undetected short and some part of the input cable, output cable or the body of the unit itself may start melting and later burning (hopefully by that moment the circuit breaker or internal fuse should have activated). The power supply may also overheat with no signs of external failure in case the protections kick in. So when you turn it on, it should be reasonably cool. You can expect same things when you connect the computer for the first time (but of course, since the PSU will have load, you can expect it to get warm). Do make a plan in advance what you're going to do if the PSU starts melting!

I thing I have covered all bad scenarios related to this (and pretty much any similar) problem. It's up to you to judge if you want to take the risk of repairing it or just get a working one. While (in my estimation) the chances of anything bad happening are very low, it's always good to be prepared.