feat: add search button to sidebar with Ctrl+K shortcut hint

Compact search bar button between sidebar header and nav. Shows
magnifying glass icon + placeholder text + ⌘K kbd hint when expanded,
just the icon when collapsed. Clicking opens the search palette.

frontend/src/lib/components/SearchPalette.svelte

@@ -17,12 +17,17 @@
 		commandTrackersCache,
 	} from '$lib/stores/caches.svelte';
 
+	let { onopen = undefined }: { onopen?: (opener: () => void) => void } = $props();
+
 	let open = $state(false);
 	let query = $state('');
 	let activeIndex = $state(0);
 	let loading = $state(false);
 	let inputEl: HTMLInputElement;
 
+	// Expose openPalette to parent via callback
+	$effect(() => { onopen?.(openPalette); });
+
 	interface SearchResult {
 		id: number;
 		name: string;

frontend/src/routes/+layout.svelte

@@ -19,6 +19,7 @@
 	const theme = getTheme();
 
 	let showPasswordForm = $state(false);
+	let openSearch: (() => void) | undefined;
 	let pwdCurrent = $state('');
 	let pwdNew = $state('');
 	let pwdMsg = $state('');
@@ -240,6 +241,22 @@
 				</button>
 			</div>
 
+			<!-- Search button -->
+			<div class="{collapsed ? 'px-2 py-1.5' : 'px-3 py-1.5'}" style="border-bottom: 1px solid var(--color-border);">
+				<button onclick={() => openSearch?.()}
+					class="flex items-center gap-2 w-full {collapsed ? 'justify-center px-2' : 'px-2.5'} py-1.5 rounded-lg text-sm transition-all duration-200"
+					style="background: var(--color-muted); color: var(--color-muted-foreground); border: 1px solid var(--color-border);"
+					onmouseenter={(e) => { e.currentTarget.style.borderColor = 'var(--color-primary)'; e.currentTarget.style.color = 'var(--color-foreground)'; }}
+					onmouseleave={(e) => { e.currentTarget.style.borderColor = 'var(--color-border)'; e.currentTarget.style.color = 'var(--color-muted-foreground)'; }}
+					title={t('searchPalette.placeholder')}>
+					<MdiIcon name="mdiMagnify" size={16} />
+					{#if !collapsed}
+						<span class="flex-1 text-left text-xs">{t('searchPalette.placeholder')}</span>
+						<kbd class="text-[0.6rem] font-mono px-1 py-0.5 rounded" style="background: var(--color-background); border: 1px solid var(--color-border);">⌘K</kbd>
+					{/if}
+				</button>
+			</div>
+
 			<!-- Nav -->
 			<nav class="flex-1 p-2 space-y-0.5 overflow-y-auto">
 				{#each navEntries as entry}
@@ -441,7 +458,7 @@
 </Modal>
 
 <Snackbar />
-<SearchPalette />
+<SearchPalette onopen={(fn) => openSearch = fn} />
 
 <style>
 	@media (max-width: 767px) {