Let's face it, Bikaner in the summer is a tough place to be. 115-120 degree F during the months of May and June, and it doesn't matter where you're living, because it is going to be hell. That being said, I can recommend looking into the Hotel Shri Ram. Yogendra Singh and Samar Rathore were great hosts, and looked after me very well during the hot season. It's only 1/2 block to a nice, quiet park for evening walks, 1 block to a cyber cafe (for recharging your phone and buying cold drinks) and about 4 blocks to the city's only ice cream parlor. If you are going to be traveling regularly to the Rajasthan State Archives, Yogendra Singh can arrive daily transport with a local autorickshawallah. I paid 20Rs. one way no matter where I went in the city.
There are at least two luxury hotels in town (Laxmi Niwas Palace and Lallgarh Palace), but they are a little removed from the city and a bit hard on the budget. Yogendra Singh can take you on a tour of Laxmi Niwas Palace, or maybe for a meal in the open air restaurant.