If you strike the deer or encounter a deer that has been struck within city limits and is obviously unable to escape, move or recover - call the local LEO entity.
Obviously, the 'city' in which this occurs will be a factor in what happens after that.
If in Omaha, let the responding officer control the actions. If he uses his firearm to dispatch the animal, there is likely a pile of paperwork for him to fill out. Once he is on scene and you advise him that you are a CHL holder or have a firearm in the vehicle and will do the deed for him, he will probably take you up on it. If he agrees, please be sure that you are exercising every safety aspect of the shoot. Protect from ricochet and fragmentation. Then you can contact the NGP and get a salvage tag, even if it is outside the seasonal limits.
In a more rural setting rules may be more lax. The local sheriff may tell you to shoot, contact NGP and remove the deer. He may also come out and do the deed himself and claim the deer for local food bank.
The use of a blade to dispatch the wounded animal would not require the calling of a local LEO unless there is need for a police accident report to be filed for vehicle damages. If you do not have damage and use your blade, you would still need to call the NGP to advise of the circumstances and disposition.