Hello

i have an RB2011UiAS-2HnD-IN and is located in my room. In our kitchen we have an Internet Radio but the Signal from my router to this room is very low.

I will expand my wireless to have perfect wireless in the kitchen or balcony or Lounge.

I have another router it is a Linksys WRT54GL so how can i achieve this?

A way to expand your wireless coverage is to use WDS - Wireless Distribution System. Basically you could configure your AP to associate as a client to an SSID, and connect other clients to itsef using the link to the main AP as a backhole.
The thing with WDS is that it's not standardized. I've used it with Linksys, and with Mikrotik routers. But I've never mixed them. So in that respect I'm not sure whether it will work between Linksys and Mikrotik. What you should do, if you haven't done it already is to flush the stock Linksys firmware and use something like DD-WRT or any other of the OpenWRT clones. These firmwares normally allow you to use WDS. But again, I'm not sure whether this will work with different vendors.

Better to use wire instead of wds and set whatever device as ap with non overlapping channel at second place closer to your kitchen. Or connect the radio directly by wire.

Connect multiple APs by wire, and configure APs on non-overlapping channels (i.e., 1, 6, 11, in the US) is always best.

However, if there is no feasible way to connect by wire, WDS works great. Just keep in mind you'll get half the bandwidth since the radio will split the time between AP mode serving clients, and WDS mode serving the backhaul. So use the 802.11ac radios (NetMetal 5) where you have triple-chain, three-stream (up to 1.3 Gbps, depending on environment) if budget allows.

Another approach is to use dual-radio devices (such as NetBox and NetMetal) so you can configure one radio dedicated to AP mode serving stations, and the other radio dedicated to the WDS (or even better, nv2 protocol).

To be clear, nv2 protocol will give you much better backhaul performance over 802.11, but you can't share it--it's for point-to-point only. So you would need two radios to use nv2 for your application; one to serve the backhaul and one to serve the stations. The WDS mode allows sharing with standard 802.11, but you only get half the bandwidth as I wrote above.

There are many options available. The best option depends on your budget, as usual.

I tried to connect to the radio directly by wire, but I have problem with authentication to ipv4. What can you advice?