Hi Liam,
The humble MS Paint will do it. Open a new file, zoom out until the canvas is all visible, choose "Paste from", browse to your scanned jpegs and choose the first one (top_half.jpg). Repeat the exercise with bottom_half.jpg and simply drag the newly-pasted image down from the top LH corner, which is where Paint will automatically insert it. Top_half.jpg will still be there, underneath it.
With practice you can align images nearly seamlessly — zoom back in closer for this, obviously. When you save your newly spliced image you may need to re-crop it subsequently (in Picasa or something) to lose the surplus white background.