Getting your phone fixed in Daegu is very easy if you can find the A/S centers. Here are A/S centers located around Daegu. There are many more around Daegu but just to make it simple I chose one per company.
First for Samsung, It is on the Yellow line, (Line number 3) at Hwanggeum station. Here is the Korean address if you want to look for it on a map yourself. [대구광역시 수성구 동대구로 110 디지털프라자 황금점 3층] The map below shows you where it is from the station.
Oddly enough there is a LG A/S center right next to the Samsung one, so the same address as the one above. Here is the map that shows you where it is from the station.
For IPhones, it is a little trickier. There are small shops all around Daegu that are certified and specialize in fixing IPhones. Here is one downtown that will be easy to find.
It is right next to the YMCA near Banwoldang.
Once you find the place it is pretty easy to fix what is broken. To fix a broken screen it will cost a minimum of 100,000 won, depends on the type of phone you have, the newer it is it will be more expensive. I hope this info is help and you can make your phone as good as new.