I agree that the room has a lot of hard surfaces which cause mid range glare and screechy highs. However streaming from the I pad leaves a lot of room for improvement. The room issues can be dealt with by strategically moving the system so it does not have to fill up a whole space. This does not mean that it won't be heard. it means that a dedicated place can have the best sound and spill over to the remainder. A preamp / Dac may be the best route. You would still stream from the Ipad but it would run to the preamp / dac. The preamp & dac will make for better sound. After that, get a dedicated streamer. The ipad is without a doubt, the weakest link. The room has good possibilities. That's why I would not be concerned with that now, especially with the budget you have
I assume you have curtains on the windows. Closing them will make for less brightness & glare. You may not notice the glare now due to low volume. but I am sure you will notice it if you turn the music volume up