I was in almost this exact situation at the beginning of last year. My dog was 12 years old at the time and seemed very healthy except for an occasional bout with diarrhea and vomiting. These symptoms persisted and so I took him to see his vet. The doctor took a couple of x-rays and discovered his spleen was enlarged. She told me it could just be that size (since there wasn't a reason for x-rays before) or she said, there could be a mass on it. She gave me the option to either have it removed or leave it alone. She also told me that "if it was her dog, she would not do the surgery". I was told to just keep an eye on him and if his symptoms got worse, they would do a biopsy and they explained the risks involved with that procedure. The spleen doesn't really "like" to be poked and prodded and if it is compromised in any way, it could rupture and the dog could bleed out into his abdomen and die. The vet also told me that it could rupture and bleed out on its own. I elected to not have the surgery and TWO WEEKS later, I came home for work and found him bloated, extremely painful, and flat out. I rushed him back to the vet, she tapped his belly and found his abdomen full of blood and they ended up doing an emergency splenectomy on him. The vet showed me see the spleen after it had been removed and told me that there was a mass on it but that the mass "didn't look like a bad guy". At that point I elected NOT to have it sent off to a lab. He recovered quickly and was back to his old self. At the beginning of this year, my current vet removed a cancerous growth from his hip. Again, everything seemed fine and he recovered.....until just a few months ago when he was diagnosed with cancer. We (I now work for the vet that performed the lump removal) did an ultrasound and found masses on his aorta. He also had a mass in his throat that appeared on the ultrasound to be compartmentalized....there is only one kind of cancer that has the same appearance and it is called hemangiosarcoma. Could the reason his spleen ruptured have been because of a hemangiosarcoma also? The lump on his hip was definitely cancerous. He would have turned 13 this coming November but I chose to put him down last month because the tumors were growing, removal and/or chemo was not an option, and his "time had come".
The reason I'm telling you all of this is in hopes that you will seriously consider doing the surgery. The vet that I work for currently said he would have insisted on taking my dog's spleen out when the x-rays showed the enlargement. And your vet is correct, a splenectomy is a rather routine procedure. The only thing I would suggest to you because of your dog's age is to have them do pre-anesthetic bloodwork to check her liver & kidney values first. That will tell them if her organs can "safely" handle the anesthesia. If money is an issue, you can always ask your vet if they would be willing to set up a payment plan.
And two other quick things....first of all, I kicked myself every day for not having his spleen removed. He clearly suffered for those hours that I was at work. Don't take that chance with your little girl. And finally, my dog's emergency surgery ended up costing me $5000+ with hospitalization, blood transfusions, etc. Believe me, he was worth EVERY last penny but I could have made a much smarter choice and saved myself money, avoided a lot of guilt and heartache, and if I would have had the spleen tested and determined it was in fact cancerous, I might have had him with me for a few more years. Money comes and goes but the gift that our pets give us lasts a lifetime.
I hope this helps and I wish you and your little girl the very best.