Barbara
We worked with a number of singers with severe pitch problems - with a degree of success in all. To my recollection it was always a case of aspiring singers not knowing how to use their vocal apparatus because they had never tried to.
In such cases we started simple ten minutes or so exercises on a song with a basic tune; with our group it was usually 'Johnny Todd' - a children's song that had been naused up by being used for the theme tune of a television police series.
The 'victim' would be asked to learn it and bring it back a week later. If they were having problems we devoted some time per session to it, usually by getting one of the group to sing a line and asking the 'struggler' to repeat the line. We persisted with this for as long as he or she was prepared to put in the time; these weren't in any way "extensive efforts at great lengths", simply persevering.
There are other 'tricks' - the much-derided 'finger-in-ear' - cupping the hand over the ear is a millennia-old technique universally used to stay in tune; as is learning to relax, tension being one of the basic causes of not being able to control pitch.      
We all learned practice pieces to improve different aspects of our singing:
Mouth music (Tail Toddle) for accuracy and breath control - a verse and a chorus in 1 breath.
A piece of Gilbert and Sullivan for diction accuracy at speed.
A Wagner piece ('Tis Ended) for handling large intervals
Another (By Evil Craft) for unusually small intervals.
Etc....
Once they are learned they remain with you, I still use them forty years on when my voice is rusty.
The voice needs to be viewed as a musical instrument if you want to sing - nobody would expect to pick up a fiddle or a concertina and play it right away, and they'd be daft to abandon the idea if it didn't work out first time round.
Jim Carroll