Well, using the monitor profile in Photoshop means the colors in your file is right on only one device on the planet, which is your monitor. To make it more universal, you need to convert the colors in your file to a device-independent profile like Adobe RGB.
In color management world, you can only assign or convert colors. Assigning doesn't change the numbers in your file, but changes the appearance of colors. Converting changes the numbers in your file, but tries to preserve the appearance of colors as much as possible. Thus, in your case, you need to convert colors to either Adobe RGB or sRGB, which is a bit more dull-looking saturation-wise, but is more stable from one device to another. And by device here I mean devices like monitors.
Now I don't know what profile is the iPad using by default to display colors onscreen, but I would assume that it is either Adobe RGB or sRGB. It could be a custom profile, but this is difficult to know. But I assume also that Apple has built support for color management in iOS, aka ColorSync.
Try either of them and let me know if it helps.
Btw, you could improve your workflow by assigning Adobe RGB to new documents from now on, as it is device-independent, and then convert them to a destination profile that matches the target device. It's just a suggestion, but a well calibrated and well profiled monitor and an Adobe RGB based workflow in Photoshop (or any other image editing app) can get you a long way.
The logic and science behind color management is extremely complex, but using it is very simple and constant. :-)