I have had a similar problem but not because I wanted access to some of the GPIO pins. You can find the pins used by the Sense Hat here. Although I am not completely sure I think the Sense Hat only needs one GND pin (please correct me if I am wrong someone). Also, try watching this tutorial by Kieth's Pi Tutorials.
To clarify (as people have disliked my post!!!), the Sense Hat uses pin 1, pin 2, pin 5 and all the GND pins or 1 GND pin (I don't know which). This means that, using a few jumper wires you could use the Sense Hat with your Raspberry Pi. My only concern is, that the Raspberry Pi Sense Hat software might not work with this set up.
Overall, it is probably best getting a new Pi and the cheapest fix is to get a Pi Zero but of course as you have not got the latest Pi I would recommend upgrading to the Raspberry Pi 3.
Hope this helps,