I've found that I often have to clear the data and cookies from the browser after using the exploit in order to get it to work again. Try that, and let us know if it changes anything. (If the browser is crashing to the Home screen, turn wifi off before opening the browser so it won't attempt and fail to load the exploit page, then you can enter the settings to delete the data/cookies. Then turn wifi back on, and go to the site. A bookmark can help make it less of a hassle.)
The 2DS can't make use of the NVRAM Profile exploit, but bluecardfix.cia should still allow you to play NDS games using the Blue Card. bluecardfix.cia should be on the system's SD card, and devmenu.3ds should be on the Gateway's mSD card. When you are in the Gateway Menu (Hold L while performing your exploit to reach it), highlight "Boot Gateway Mode" and hold B. The text will change to "Boot Gateway Sysnand Mode". While holding B, press A to select that, and it will ask to confirm. Do so, and then run DevMenu like you would any other game backup. (Technically, if you still haven't set up emuNand, you don't really have to go to the menu, because if there's no emuNand, it will default to booting sysNand in Gateway Mode.) Use DevMenu to install bluecardfix.cia to your sysNand. You will need to restart your system after that, because the Blue Card (and any other NDS-mode software) will not run in Gateway Mode or in emuNand.
Edit: Opened the thread a while ago, got distracted, forgot to check for new posts before coming back. Saw that you said you've tried clearing your cookies/history. If you've done that, and none of the mirrors are working for you, then I'm a little out of ideas as to why you can't reach the Launcher. Sorry about that.