If I understand this right, you want to play staccato RH, and legato LH? If that's it, I can help. If not, please help me clarify what I'm misunderstanding.
That is right. How do you do it?
To start, as simply as possible.
This is nothing short of a technique thing -- something you have to learn in your hands before you can apply it to a piece of music. Once you learn it, you won't need to relearn it for other pieces of music. It will stay with you. But it's one of the bigger pains to develop, and really requires 100% focus and active participation on the part of your brain until you get the hang of it.
Try it with one note. Repeat a note in the LH (with only one finger), while repeating a note in the RH (with only one finger). Start playing them the same way. Slowly make one of them more staccato. Then, return to normal. Then make the other finger staccato. Then, return no normal. Then, practice switching on the fly. Then try switching every other note, or every couple of notes at random until you get full control over it.
Next, try this with a five-note pattern. Notice, we're slowly upping the difficulty here. No crossovers, no big changes. No leaping. Just five successive notes.
If you're stuck, sometimes picking up the staccato hand off the keys helps to develop that staccato sound. Eventually, you'll be able to control it with a slightly smaller movement, but you're still going to "bounce" just a little bit to get that staccato sound. (I put bounce in quotes because it's a misnomer.. I just can't think of the right word off the top of my head.)
Next, go to a scale. Try arpeggios. Whatever helps you facilitate the movements. If you feel like you've really got the hang of it, go back to the piece and give it a go.