hello there
I think she is a bit young for controlled crying at five months, she wouldn't get. They have to be wise enough to realise that you mean business and that of the principle of cause and effect ans she is too young for that, she barely knows that you and she aren't one entity. I think 6 or 7 months would be a good time to introduce Controlled Crying (CC)
Hate to tell you but people whose small babies sleep through the night usually can't believe their luck and it may be a phase that gets disrupted when the teeth come through. Don't mean to sound negative, just want to give you a realistic picture.
But what you can do is get her into a routine that she will associate with settling herself. Take it step by step, do bath time and singing or what ever your evening routine is and then lay her in her cot with lots of strokes and resassuring noises and hold the dummy for her but just don't pick her up, that will be a step in the right direction.
In my opinion, she is too small to get through the night without a feed, are you breast or bottle feeding her? Again the general consensus is that 6 months is the earliest you can reasonalby expect then to do that.
I can understand how tiring and frustrating it is not to have a child that sleeps well (don't get me started on my two boys) but at the moment the biggest help you can be to her and yourself is just taking things step by step in the direction of encouraging her to settle with you present but not holding her, then eventually just by you placing her in her cot and patting her and singing to her.
Hope this helps
By the way, where are you in Australia? I'm in Perth (or are you in Austria?)